2011

• Paper 1, Section I, $3 F$

  #Analysis I

  2011

  (a) State, without proof, the Bolzano-Weierstrass Theorem.

  (b) Give an example of a sequence that does not have a convergent subsequence.

  (c) Give an example of an unbounded sequence having a convergent subsequence.

  (d) Let $a_{n}=1+(-1)^{\lfloor n / 2\rfloor}(1+1 / n)$, where $\lfloor x\rfloor$ denotes the integer part of $x$. Find all values $c$ such that the sequence $\left\{a_{n}\right\}$ has a subsequence converging to $c$. For each such value, provide a subsequence converging to it.

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• Paper 1, Section I, D

  #Analysis I

  2011

  Find the radius of convergence of each of the following power series. (i) $\sum_{n \geqslant 1} n^{2} z^{n}$ (ii) $\sum_{n \geqslant 1} n^{n^{1 / 3}} z^{n}$

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• Paper 1, Section II, F

  #Analysis I

  2011

  (a) State, without proof, the ratio test for the series $\sum_{n \geqslant 1} a_{n}$, where $a_{n}>0$. Give examples, without proof, to show that, when $a_{n+1}<a_{n}$ and $a_{n+1} / a_{n} \rightarrow 1$, the series may converge or diverge.

  (b) Prove that $\sum_{k=1}^{n-1} \frac{1}{k} \geqslant \log n$.

  (c) Now suppose that $a_{n}>0$ and that, for $n$ large enough, $\frac{a_{n+1}}{a_{n}} \leqslant 1-\frac{c}{n}$ where $c>1$. Prove that the series $\sum_{n \geqslant 1} a_{n}$ converges.

  [You may find it helpful to prove the inequality $\log (1-x)<-x$ for $0<x<1$.]

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• Paper 1, Section II, E

  #Analysis I

  2011

  State and prove the Intermediate Value Theorem.

  A fixed point of a function $f: X \rightarrow X$ is an $x \in X$ with $f(x)=x$. Prove that every continuous function $f:[0,1] \rightarrow[0,1]$ has a fixed point.

  Answer the following questions with justification.

  (i) Does every continuous function $f:(0,1) \rightarrow(0,1)$ have a fixed point?

  (ii) Does every continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ have a fixed point?

  (iii) Does every function $f:[0,1] \rightarrow[0,1]$ (not necessarily continuous) have a fixed point?

  (iv) Let $f:[0,1] \rightarrow[0,1]$ be a continuous function with $f(0)=1$ and $f(1)=0$. Can $f$ have exactly two fixed points?

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• Paper 1, Section II, E

  #Analysis I

  2011

  For each of the following two functions $f: \mathbb{R} \rightarrow \mathbb{R}$, determine the set of points at which $f$ is continuous, and also the set of points at which $f$ is differentiable.

  $$\begin{aligned} &\text { (i) } f(x)= \begin{cases}x & \text { if } x \in \mathbb{Q} \\ -x & \text { if } x \notin \mathbb{Q}\end{cases} \\ &\text { (ii) } f(x)= \begin{cases}x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} \end{aligned}$$

  By modifying the function in (i), or otherwise, find a function (not necessarily continuous) $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is differentiable at 0 and nowhere else.

  Find a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is not differentiable at the points $1 / 2,1 / 3,1 / 4, \ldots$, but is differentiable at all other points.

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• Paper 1, Section II, D

  #Analysis I

  2011

  State and prove the Fundamental Theorem of Calculus.

  Let $f:[0,1] \rightarrow \mathbb{R}$ be integrable, and set $F(x)=\int_{0}^{x} f(t) \mathrm{d} t$ for $0<x<1$. Must $F$ be differentiable?

  Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must the Riemann integral of $g$ from 0 to 1 exist?

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• Paper 2, Section I, A

  #Differential Equations

  2011

  (a) Consider the homogeneous $k$ th-order difference equation

  $$a_{k} y_{n+k}+a_{k-1} y_{n+k-1}+\ldots+a_{2} y_{n+2}+a_{1} y_{n+1}+a_{0} y_{n}=0$$

  where the coefficients $a_{k}, \ldots, a_{0}$ are constants. Show that for $\lambda \neq 0$ the sequence $y_{n}=\lambda^{n}$ is a solution if and only if $p(\lambda)=0$, where

  $$p(\lambda)=a_{k} \lambda^{k}+a_{k-1} \lambda^{k-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}$$

  State the general solution of $(*)$ if $k=3$ and $p(\lambda)=(\lambda-\mu)^{3}$ for some constant $\mu$.

  (b) Find an inhomogeneous difference equation that has the general solution

  $$y_{n}=a 2^{n}-n, \quad a \in \mathbb{R}$$

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• Paper 2, Section I, $2 A$

  #Differential Equations

  2011

  (a) For a differential equation of the form $\frac{\mathrm{d} y}{\mathrm{~d} x}=f(y)$, explain how $f^{\prime}(y)$ can be used to determine the stability of any equilibrium solutions and justify your answer.

  (b) Find the equilibrium solutions of the differential equation

  $$\frac{\mathrm{d} y}{\mathrm{~d} x}=y^{3}-y^{2}-2 y$$

  and determine their stability. Sketch representative solution curves in the $(x, y)$-plane.

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• Paper 2, Section II, A

  #Differential Equations

  2011

  (a) Find the general real solution of the system of first-order differential equations

  $$\begin{aligned} &\dot{x}=x+\mu y \\ &\dot{y}=-\mu x+y, \end{aligned}$$

  where $\mu$ is a real constant.

  (b) Find the fixed points of the non-linear system of first-order differential equations

  $$\begin{aligned} &\dot{x}=x+y \\ &\dot{y}=-x+y-2 x^{2} y \end{aligned}$$

  and determine their nature. Sketch the phase portrait indicating the direction of motion along trajectories.

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• Paper 2, Section II, A

  #Differential Equations

  2011

  (a) A surface in $\mathbb{R}^{3}$ is defined by the equation $f(x, y, z)=c$, where $c$ is a constant. Show that the partial derivatives on this surface satisfy

  $$\left.\left.\left.\frac{\partial x}{\partial y}\right|_{z} \frac{\partial y}{\partial z}\right|_{x} \frac{\partial z}{\partial x}\right|_{y}=-1$$

  (b) Now let $f(x, y, z)=x^{2}-y^{4}+2 a y^{2}+z^{2}$, where $a$ is a constant.

  (i) Find expressions for the three partial derivatives $\left.\frac{\partial x}{\partial y}\right|_{z},\left.\frac{\partial y}{\partial z}\right|_{x}$ and $\left.\frac{\partial z}{\partial x}\right|_{y}$ on the surface $f(x, y, z)=c$, and verify the identity $(*)$.

  (ii) Find the rate of change of $f$ in the radial direction at the point $(x, 0, z)$.

  (iii) Find and classify the stationary points of $f$.

  (iv) Sketch contour plots of $f$ in the $(x, y)$-plane for the cases $a=1$ and $a=-1$.

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• Paper 2, Section II, $7 \mathrm{~A}$

  #Differential Equations

  2011

  (a) Define the Wronskian $W$ of two solutions $y_{1}(x)$ and $y_{2}(x)$ of the differential equation

  $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

  and state a necessary and sufficient condition for $y_{1}(x)$ and $y_{2}(x)$ to be linearly independent. Show that $W(x)$ satisfies the differential equation

  $$W^{\prime}(x)=-p(x) W(x)$$

  (b) By evaluating the Wronskian, or otherwise, find functions $p(x)$ and $q(x)$ such that $(*)$ has solutions $y_{1}(x)=1+\cos x$ and $y_{2}(x)=\sin x$. What is the value of $W(\pi) ?$ Is there a unique solution to the differential equation for $0 \leqslant x<\infty$ with initial conditions $y(0)=0, y^{\prime}(0)=1$ ? Why or why not?

  (c) Write down a third-order differential equation with constant coefficients, such that $y_{1}(x)=1+\cos x$ and $y_{2}(x)=\sin x$ are both solutions. Is the solution to this equation for $0 \leqslant x<\infty$ with initial conditions $y(0)=y^{\prime \prime}(0)=0, y^{\prime}(0)=1$ unique? Why or why not?

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• Paper 2, Section II, A

  #Differential Equations

  2011

  (a) The circumference $y$ of an ellipse with semi-axes 1 and $x$ is given by

  $$y(x)=\int_{0}^{2 \pi} \sqrt{\sin ^{2} \theta+x^{2} \cos ^{2} \theta} \mathrm{d} \theta .$$

  Setting $t=1-x^{2}$, find the first three terms in a series expansion of $(*)$ around $t=0$.

  (b) Euler proved that $y$ also satisfies the differential equation

  $$x\left(1-x^{2}\right) y^{\prime \prime}-\left(1+x^{2}\right) y^{\prime}+x y=0 .$$

  Use the substitution $t=1-x^{2}$ for $x \geqslant 0$ to find a differential equation for $u(t)$, where $u(t)=y(x)$. Show that this differential equation has regular singular points at $t=0$ and $t=1$.

  Show that the indicial equation at $t=0$ has a repeated root, and find the recurrence relation for the coefficients of the corresponding power-series solution. State the form of a second, independent solution.

  Verify that the power-series solution is consistent with your answer in (a).

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• Paper 4, Section I, B

  #Dynamics and Relativity

  2011

  The motion of a planet in the gravitational field of a star of mass $M$ obeys

  $$\frac{\mathrm{d}^{2} r}{\mathrm{~d} t^{2}}-\frac{h^{2}}{r^{3}}=-\frac{G M}{r^{2}}, \quad r^{2} \frac{\mathrm{d} \theta}{\mathrm{d} t}=h$$

  where $r(t)$ and $\theta(t)$ are polar coordinates in a plane and $h$ is a constant. Explain one of Kepler's Laws by giving a geometrical interpretation of $h$.

  Show that circular orbits are possible, and derive another of Kepler's Laws relating the radius $a$ and the period $T$ of such an orbit. Show that any circular orbit is stable under small perturbations that leave $h$ unchanged.

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• Paper 4, Section I, B

  #Dynamics and Relativity

  2011

  Inertial frames $S$ and $S^{\prime}$ in two-dimensional space-time have coordinates $(x, t)$ and $\left(x^{\prime}, t^{\prime}\right)$, respectively. These coordinates are related by a Lorentz transformation with $v$ the velocity of $S^{\prime}$ relative to $S$. Show that if $x_{\pm}=x \pm c t$ and $x_{\pm}^{\prime}=x^{\prime} \pm c t^{\prime}$ then the Lorentz transformation can be expressed in the form

  $$x_{+}^{\prime}=\lambda(v) x_{+} \quad \text { and } \quad x_{-}^{\prime}=\lambda(-v) x_{-}, \quad \text { where } \quad \lambda(v)=\left(\frac{c-v}{c+v}\right)^{1 / 2} .$$

  Deduce that $x^{2}-c^{2} t^{2}=x^{\prime 2}-c^{2} t^{\prime 2}$.

  Use the form $(*)$ to verify that successive Lorentz transformations with velocities $v_{1}$ and $v_{2}$ result in another Lorentz transformation with velocity $v_{3}$, to be determined in terms of $v_{1}$ and $v_{2}$.

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2011

  A particle with mass $m$ and position $\mathbf{r}(t)$ is subject to a force

  $$\mathbf{F}=\mathbf{A}(\mathbf{r})+\dot{\mathbf{r}} \times \mathbf{B}(\mathbf{r})$$

  (a) Suppose that $\mathbf{A}=-\nabla \phi$. Show that

  $$E=\frac{1}{2} m \dot{\mathbf{r}}^{2}+\phi(\mathbf{r})$$

  is constant, and interpret this result, explaining why the field $\mathbf{B}$ plays no role.

  (b) Suppose, in addition, that $\mathbf{B}=-\nabla \psi$ and that both $\phi$ and $\psi$ depend only on $r=|\mathbf{r}|$. Show that

  $$\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}-\psi \mathbf{r}$$

  is independent of time if $\psi(r)=\mu / r$, for any constant $\mu$.

  (c) Now specialise further to the case $\psi=0$. Explain why the result in (b) implies that the motion of the particle is confined to a plane. Show also that

  $$\mathbf{K}=\mathbf{L} \times \dot{\mathbf{r}}-\phi \mathbf{r}$$

  is constant provided $\phi(r)$ takes a certain form, to be determined.

  [ Recall that $\mathbf{r} \cdot \dot{\mathbf{r}}=r \dot{r}$ and that if $f$ depends only on $r=|\mathbf{r}|$ then $\left.\boldsymbol{\nabla} f=f^{\prime}(r) \hat{\mathbf{r}} .\right]$

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2011

  The trajectory of a particle $\mathbf{r}(t)$ is observed in a frame $S$ which rotates with constant angular velocity $\omega$ relative to an inertial frame $I$. Given that the time derivative in $I$ of any vector $\mathbf{u}$ is

  $$\left(\frac{\mathrm{d} \mathbf{u}}{\mathrm{d} t}\right)_{I}=\dot{\mathbf{u}}+\boldsymbol{\omega} \times \mathbf{u},$$

  where a dot denotes a time derivative in $S$, show that

  $$m \ddot{\mathbf{r}}=\mathbf{F}-2 m \boldsymbol{\omega} \times \dot{\mathbf{r}}-m \boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}),$$

  where $\mathbf{F}$ is the force on the particle and $m$ is its mass.

  Let $S$ be the frame that rotates with the Earth. Assume that the Earth is a sphere of radius $R$. Let $P$ be a point on its surface at latitude $\pi / 2-\theta$, and define vertical to be the direction normal to the Earth's surface at $P$.

  (a) A particle at $P$ is released from rest in $S$ and is acted on only by gravity. Show that its initial acceleration makes an angle with the vertical of approximately

  $$\frac{\omega^{2} R}{g} \sin \theta \cos \theta$$

  working to lowest non-trivial order in $\omega$.

  (b) Now consider a particle fired vertically upwards from $P$ with speed $v$. Assuming that terms of order $\omega^{2}$ and higher can be neglected, show that it falls back to Earth under gravity at a distance

  $$\frac{4}{3} \frac{\omega v^{3}}{g^{2}} \sin \theta$$

  from $P$. [You may neglect the curvature of the Earth's surface and the vertical variation of gravity.]

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2011

  A rocket carries equipment to collect samples from a stationary cloud of cosmic dust. The rocket moves in a straight line, burning fuel and ejecting gas at constant speed $u$ relative to itself. Let $v(t)$ be the speed of the rocket, $M(t)$ its total mass, including fuel and any dust collected, and $m(t)$ the total mass of gas that has been ejected. Show that

  $$M \frac{\mathrm{d} v}{\mathrm{~d} t}+v \frac{\mathrm{d} M}{\mathrm{~d} t}+(v-u) \frac{\mathrm{d} m}{\mathrm{dt}}=0$$

  assuming that all external forces are negligible.

  (a) If no dust is collected and the rocket starts from rest with mass $M_{0}$, deduce that

  $$v=u \log \left(M_{0} / M\right)$$

  (b) If cosmic dust is collected at a constant rate of $\alpha$ units of mass per unit time and fuel is consumed at a constant rate $\mathrm{d} m / \mathrm{d} t=\beta$, show that, with the same initial conditions as in (a),

  $$v=\frac{u \beta}{\alpha}\left(1-\left(M / M_{0}\right)^{\alpha /(\beta-\alpha)}\right)$$

  Verify that the solution in (a) is recovered in the limit $\alpha \rightarrow 0$.

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2011

  (a) Write down the relativistic energy $E$ of a particle of rest mass $m$ and speed $v$. Find the approximate form for $E$ when $v$ is small compared to $c$, keeping all terms up to order $(v / c)^{2}$. What new physical idea (when compared to Newtonian Dynamics) is revealed in this approximation?

  (b) A particle of rest mass $m$ is fired at an identical particle which is at rest in the laboratory frame. Let $E$ be the relativistic energy and $v$ the speed of the incident particle in this frame. After the collision, there are $N$ particles in total, each with rest mass $m$. Assuming that four-momentum is conserved, find a lower bound on $E$ and hence show that

  $$v \geqslant \frac{N\left(N^{2}-4\right)^{1 / 2}}{N^{2}-2} c$$

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• Paper 3, Section I, D

  #Groups

  2011

  (a) Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.

  (b) The symmetric group $S_{n}$ acts on the set $X=\{1, \ldots, n\}$, and hence acts on $X \times X$ by $g(x, y)=(g x, g y)$. Determine the orbits of $S_{n}$ on $X \times X$.

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• Paper 3, Section I, D

  #Groups

  2011

  State and prove Lagrange's Theorem.

  Show that the dihedral group of order $2 n$ has a subgroup of order $k$ for every $k$ dividing $2 n$.

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• Paper 3, Section II, D

  #Groups

  2011

  (a) Let $G$ be a finite group, and let $g \in G$. Define the order of $g$ and show it is finite. Show that if $g$ is conjugate to $h$, then $g$ and $h$ have the same order.

  (b) Show that every $g \in S_{n}$ can be written as a product of disjoint cycles. For $g \in S_{n}$, describe the order of $g$ in terms of the cycle decomposition of $g$.

  (c) Define the alternating group $A_{n}$. What is the condition on the cycle decomposition of $g \in S_{n}$ that characterises when $g \in A_{n}$ ?

  (d) Show that, for every $n, A_{n+2}$ has a subgroup isomorphic to $S_{n}$.

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• Paper 3, Section II, D

  #Groups

  2011

  (a) Let

  $$S L_{2}(\mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{Z}\right\}$$

  and, for a prime $p$, let

  $$S L_{2}\left(\mathbb{F}_{p}\right)=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{F}_{p}\right\}$$

  where $\mathbb{F}_{p}$ consists of the elements $0,1, \ldots, p-1$, with addition and multiplication mod $p$.

  Show that $S L_{2}(\mathbb{Z})$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ are groups under matrix multiplication.

  [You may assume that matrix multiplication is associative, and that the determinant of a product equals the product of the determinants.]

  By defining a suitable homomorphism from $S L_{2}(\mathbb{Z}) \rightarrow S L_{2}\left(\mathbb{F}_{5}\right)$, show that

  $$\left\{\left(\begin{array}{cc} 1+5 a & 5 b \\ 5 c & 1+5 d \end{array}\right) \in S L_{2}(\mathbb{Z}) \mid a, b, c, d \in \mathbb{Z}\right\}$$

  is a normal subgroup of $S L_{2}(\mathbb{Z})$.

  (b) Define the group $G L_{2}\left(\mathbb{F}_{5}\right)$, and show that it has order 480 . By defining a suitable homomorphism from $G L_{2}\left(\mathbb{F}_{5}\right)$ to another group, which should be specified, show that the order of $S L_{2}\left(\mathbb{F}_{5}\right)$ is 120 .

  Find a subgroup of $G L_{2}\left(\mathbb{F}_{5}\right)$ of index 2 .

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• Paper 3, Section II, $7 \mathrm{D}$

  #Groups

  2011

  (a) State the orbit-stabilizer theorem.

  Let a group $G$ act on itself by conjugation. Define the centre $Z(G)$ of $G$, and show that $Z(G)$ consists of the orbits of size 1 . Show that $Z(G)$ is a normal subgroup of $G$.

  (b) Now let $|G|=p^{n}$, where $p$ is a prime and $n \geqslant 1$. Show that if $G$ acts on a set $X$, and $Y$ is an orbit of this action, then either $|Y|=1$ or $p$ divides $|Y|$.

  Show that $|Z(G)|>1$.

  By considering the set of elements of $G$ that commute with a fixed element $x$ not in $Z(G)$, show that $Z(G)$ cannot have order $p^{n-1}$.

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• Paper 3, Section II, D

  #Groups

  2011

  (a) Let $G$ be a finite group and let $H$ be a subgroup of $G$. Show that if $|G|=2|H|$ then $H$ is normal in $G$.

  Show that the dihedral group $D_{2 n}$ of order $2 n$ has a normal subgroup different from both $D_{2 n}$ and $\{e\}$.

  For each integer $k \geqslant 3$, give an example of a finite group $G$, and a subgroup $H$, such that $|G|=k|H|$ and $H$ is not normal in $G$.

  (b) Show that $A_{5}$ is a simple group.

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• Paper 4, Section I, E

  #Numbers and Sets

  2011

  What does it mean to say that a function $f: X \rightarrow Y$ has an inverse? Show that a function has an inverse if and only if it is a bijection.

  Let $f$ and $g$ be functions from a set $X$ to itself. Which of the following are always true, and which can be false? Give proofs or counterexamples as appropriate.

  (i) If $f$ and $g$ are bijections then $f \circ g$ is a bijection.

  (ii) If $f \circ g$ is a bijection then $f$ and $g$ are bijections.

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• Paper 4, Section I, E

  #Numbers and Sets

  2011

  What is an equivalence relation on a set $X ?$ If $\sim$ is an equivalence relation on $X$, what is an equivalence class of $\sim$ ? Prove that the equivalence classes of $\sim$ form a partition of $X$.

  Let $\sim$ be the relation on the positive integers defined by $x \sim y$ if either $x$ divides $y$ or $y$ divides $x$. Is $\sim$ an equivalence relation? Justify your answer.

  Write down an equivalence relation on the positive integers that has exactly four equivalence classes, of which two are infinite and two are finite.

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• Paper 4, Section II, E

  #Numbers and Sets

  2011

  (a) What is the highest common factor of two positive integers $a$ and $b$ ? Show that the highest common factor may always be expressed in the form $\lambda a+\mu b$, where $\lambda$ and $\mu$ are integers.

  Which positive integers $n$ have the property that, for any positive integers $a$ and $b$, if $n$ divides $a b$ then $n$ divides $a$ or $n$ divides $b$ ? Justify your answer.

  Let $a, b, c, d$ be distinct prime numbers. Explain carefully why $a b$ cannot equal $c d$.

  [No form of the Fundamental Theorem of Arithmetic may be assumed without proof.]

  (b) Now let $S$ be the set of positive integers that are congruent to 1 mod 10 . We say that $x \in S$ is irreducible if $x>1$ and whenever $a, b \in S$ satisfy $a b=x$ then $a=1$ or $b=1$. Do there exist distinct irreducibles $a, b, c, d$ with $a b=c d ?$

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• Paper 4, Section II, E

  #Numbers and Sets

  2011

  State Fermat's Theorem and Wilson's Theorem.

  Let $p$ be a prime.

  (a) Show that if $p \equiv 3(\bmod 4)$ then the equation $x^{2} \equiv-1(\bmod p)$ has no solution.

  (b) By considering $\left(\frac{p-1}{2}\right)$ !, or otherwise, show that if $p \equiv 1(\bmod 4)$ then the equation $x^{2} \equiv-1(\bmod p)$ does have a solution.

  (c) Show that if $p \equiv 2(\bmod 3)$ then the equation $x^{3} \equiv-1(\bmod p)$ has no solution other than $-1(\bmod p)$.

  (d) Using the fact that $14^{2} \equiv-3(\bmod 199)$, find a solution of $x^{3} \equiv-1(\bmod 199)$ that is not $-1(\bmod 199)$.

  [Hint: how are the complex numbers $\sqrt{-3}$ and $\sqrt[3]{-1}$ related?]

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• Paper 4, Section II, $7 \mathrm{E}$

  #Numbers and Sets

  2011

  Define the binomial coefficient $\left(\begin{array}{l}n \\ i\end{array}\right)$, where $n$ is a positive integer and $i$ is an integer with $0 \leqslant i \leqslant n$. Arguing from your definition, show that $\sum_{i=0}^{n}\left(\begin{array}{l}n \\ i\end{array}\right)=2^{n}$.

  Prove the binomial theorem, that $(1+x)^{n}=\sum_{i=0}^{n}\left(\begin{array}{l}n \\ i\end{array}\right) x^{i}$ for any real number $x$.

  By differentiating this expression, or otherwise, evaluate $\sum_{i=0}^{n} i\left(\begin{array}{l}n \\ i\end{array}\right)$ and $\sum_{i=0}^{n} i^{2}\left(\begin{array}{l}n \\ i\end{array}\right)$. By considering the identity $(1+x)^{n}(1+x)^{n}=(1+x)^{2 n}$, or otherwise, show that

  $$\sum_{i=0}^{n}\left(\begin{array}{l} n \\ i \end{array}\right)^{2}=\left(\begin{array}{c} 2 n \\ n \end{array}\right)$$

  Show that $\sum_{i=0}^{n} i\left(\begin{array}{c}n \\ i\end{array}\right)^{2}=\frac{n}{2}\left(\begin{array}{c}2 n \\ n\end{array}\right)$

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• Paper 4, Section II, E

  #Numbers and Sets

  2011

  Show that, for any set $X$, there is no surjection from $X$ to the power-set of $X$.

  Show that there exists an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

  Let $A$ be a subset of $\mathbb{R}^{2}$. A section of $A$ is a subset of $\mathbb{R}$ of the form

  $$\{t \in \mathbb{R}: a+t b \in A\},$$

  where $a \in \mathbb{R}^{2}$ and $b \in \mathbb{R}^{2}$ with $b \neq 0$. Prove that there does not exist a set $A \subset \mathbb{R}^{2}$ such that every set $S \subset \mathbb{R}$ is a section of $A$.

  Does there exist a set $A \subset \mathbb{R}^{2}$ such that every countable set $S \subset \mathbb{R}$ is a section of $A ? \quad$ [There is no requirement that every section of $A$ should be countable.] Justify your answer.

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• Paper 2, Section I, F

  #Probability

  2011

  Let $X$ be a random variable taking non-negative integer values and let $Y$ be a random variable taking real values.

  (a) Define the probability-generating function $G_{X}(s)$. Calculate it explicitly for a Poisson random variable with mean $\lambda>0$.

  (b) Define the moment-generating function $M_{Y}(t)$. Calculate it explicitly for a normal random variable $\mathrm{N}(0,1)$.

  (c) By considering a random sum of independent copies of $Y$, prove that, for general $X$ and $Y, G_{X}\left(M_{Y}(t)\right)$ is the moment-generating function of some random variable.

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• Paper 2, Section I, F

  #Probability

  2011

  What does it mean to say that events $A_{1}, \ldots, A_{n}$ are (i) pairwise independent, (ii) independent?

  Consider pairwise disjoint events $B_{1}, B_{2}, B_{3}$ and $C$, with

  $$\mathbb{P}\left(B_{1}\right)=\mathbb{P}\left(B_{2}\right)=\mathbb{P}\left(B_{3}\right)=p \text { and } \mathbb{P}(C)=q, \text { where } 3 p+q \leqslant 1$$

  Let $0 \leqslant q \leqslant 1 / 16$. Prove that the events $B_{1} \cup C, B_{2} \cup C$ and $B_{3} \cup C$ are pairwise independent if and only if

  $$p=-q+\sqrt{q} .$$

  Prove or disprove that there exist $p>0$ and $q>0$ such that these three events are independent.

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• Paper 2, Section II, F

  #Probability

  2011

  (a) Let $B_{1}, \ldots, B_{n}$ be pairwise disjoint events such that their union $B_{1} \cup B_{2} \cup \ldots \cup B_{n}$ gives the whole set of outcomes, with $\mathbb{P}\left(B_{i}\right)>0$ for $1 \leqslant i \leqslant n$. Prove that for any event $A$ with $\mathbb{P}(A)>0$ and for any $i$

  $$\mathbb{P}\left(B_{i} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)}{\sum_{1 \leqslant j \leqslant n} \mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}$$

  (b) A prince is equally likely to sleep on any number of mattresses from six to eight; on half the nights a pea is placed beneath the lowest mattress. With only six mattresses his sleep is always disturbed by the presence of a pea; with seven a pea, if present, is unnoticed in one night out of five; and with eight his sleep is undisturbed despite an offending pea in two nights out of five.

  What is the probability that, on a given night, the prince's sleep was undisturbed?

  On the morning of his wedding day, he announces that he has just spent the most peaceful and undisturbed of nights. What is the expected number of mattresses on which he slept the previous night?

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• Paper 2, Section II, F

  #Probability

  2011

  (a) State Markov's inequality.

  (b) Let $r$ be a given positive integer. You toss an unbiased coin repeatedly until the first head appears, which occurs on the $H_{1}$ th toss. Next, I toss the same coin until I get my first tail, which occurs on my $T_{1}$ th toss. Then you continue until you get your second head with a further $H_{2}$ tosses; then I continue with a further $T_{2}$ tosses until my second tail. We continue for $r$ turns like this, and generate a sequence $H_{1}, T_{1}, H_{2}, T_{2}, \ldots, H_{r}$, $T_{r}$ of random variables. The total number of tosses made is $Y_{r}$. (For example, for $r=2$, a sequence of outcomes $t t h|t| t t t h \mid h h t$ gives $H_{1}=3, T_{1}=1, H_{2}=4, T_{2}=3$ and $Y_{2}=11$.)

  Find the probability-generating functions of the random variables $H_{j}$ and $T_{j}$. Hence or otherwise obtain the mean values $\mathbb{E} H_{j}$ and $\mathbb{E} T_{j}$.

  Obtain the probability-generating function of the random variable $Y_{r}$, and find the mean value $\mathbb{E} Y_{r}$.

  Prove that, for $n \geqslant 2 r$,

  $$\mathbb{P}\left(Y_{r}=n\right)=\frac{1}{2^{n}}\left(\begin{array}{c} n-1 \\ 2 r-1 \end{array}\right)$$

  For $r=1$, calculate $\mathbb{P}\left(Y_{1} \geqslant 5\right)$, and confirm that it satisfies Markov's inequality.

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• Paper 2, Section II, F

  #Probability

  2011

  I was given a clockwork orange for my birthday. Initially, I place it at the centre of my dining table, which happens to be exactly 20 units long. One minute after I place it on the table it moves one unit towards the left end of the table or one unit towards the right, each with probability 1/2. It continues in this manner at one minute intervals, with the direction of each move being independent of what has gone before, until it reaches either end of the table where it promptly falls off. If it falls off the left end it will break my Ming vase. If it falls off the right end it will land in a bucket of sand leaving the vase intact.

  (a) Derive the difference equation for the probability that the Ming vase will survive, in terms of the current distance $k$ from the orange to the left end, where $k=1, \ldots, 19$.

  (b) Derive the corresponding difference equation for the expected time when the orange falls off the table.

  (c) Write down the general formula for the solution of each of the difference equations from (a) and (b). [No proof is required.]

  (d) Based on parts (a)-(c), calculate the probability that the Ming vase will survive if, instead of placing the orange at the centre of the table, I place it initially 3 units from the right end of the table. Calculate the expected time until the orange falls off.

  (e) Suppose I place the orange 3 units from the left end of the table. Calculate the probability that the orange will fall off the right end before it reaches a distance 1 unit from the left end of the table.

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• Paper 2, Section II, F

  #Probability

  2011

  A circular island has a volcano at its central point. During an eruption, lava flows from the mouth of the volcano and covers a sector with random angle $\Phi$ (measured in radians), whose line of symmetry makes a random angle $\Theta$ with some fixed compass bearing.

  The variables $\Theta$ and $\Phi$ are independent. The probability density function of $\Theta$ is constant on $(0,2 \pi)$ and the probability density function of $\Phi$ is of the form $A(\pi-\phi / 2)$ where $0<\phi<2 \pi$, and $A$ is a constant.

  (a) Find the value of $A$. Calculate the expected value and the variance of the sector angle $\Phi$. Explain briefly how you would simulate the random variable $\Phi$ using a uniformly distributed random variable $U$.

  (b) $H_{1}$ and $H_{2}$ are two houses on the island which are collinear with the mouth of the volcano, but on different sides of it. Find

  (i) the probability that $H_{1}$ is hit by the lava;

  (ii) the probability that both $H_{1}$ and $H_{2}$ are hit by the lava;

  (iii) the probability that $H_{2}$ is not hit by the lava given that $H_{1}$ is hit.

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• Paper 3, Section I, C

  #Vector Calculus

  2011

  Cartesian coordinates $x, y, z$ and spherical polar coordinates $r, \theta, \phi$ are related by

  $$x=r \sin \theta \cos \phi, \quad y=r \sin \theta \sin \phi, \quad z=r \cos \theta$$

  Find scalars $h_{r}, h_{\theta}, h_{\phi}$ and unit vectors $\mathbf{e}_{r}, \mathbf{e}_{\theta}, \mathbf{e}_{\phi}$ such that

  $$\mathrm{d} \mathbf{x}=h_{r} \mathbf{e}_{r} \mathrm{~d} r+h_{\theta} \mathbf{e}_{\theta} \mathrm{d} \theta+h_{\phi} \mathbf{e}_{\phi} \mathrm{d} \phi$$

  Verify that the unit vectors are mutually orthogonal.

  Hence calculate the area of the open surface defined by $\theta=\alpha, 0 \leqslant r \leqslant R$, $0 \leqslant \phi \leqslant 2 \pi$, where $\alpha$ and $R$ are constants.

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• Paper 3, Section I, C

  #Vector Calculus

  2011

  State the value of $\partial x_{i} / \partial x_{j}$ and find $\partial r / \partial x_{j}$, where $r=|\mathbf{x}|$.

  Vector fields $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{3}$ are given by $\mathbf{u}=r^{\alpha} \mathbf{x}$ and $\mathbf{v}=\mathbf{k} \times \mathbf{u}$, where $\alpha$ is a constant and $\mathbf{k}$ is a constant vector. Calculate the second-rank tensor $d_{i j}=\partial u_{i} / \partial x_{j}$, and deduce that $\boldsymbol{\nabla} \times \mathbf{u}=\mathbf{0}$ and $\boldsymbol{\nabla} \cdot \mathbf{v}=0$. When $\alpha=-3$, show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and

  $$\nabla \times \mathbf{v}=\frac{3(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}-\mathbf{k} r^{2}}{r^{5}}$$

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• Paper 3, Section II, C

  #Vector Calculus

  2011

  Write down the most general isotropic tensors of rank 2 and 3. Use the tensor transformation law to show that they are, indeed, isotropic.

  Let $V$ be the sphere $0 \leqslant r \leqslant a$. Explain briefly why

  $$T_{i_{1} \ldots i_{n}}=\int_{V} x_{i_{1}} \ldots x_{i_{n}} \mathrm{~d} V$$

  is an isotropic tensor for any $n$. Hence show that

  $$\int_{V} x_{i} x_{j} \mathrm{~d} V=\alpha \delta_{i j}, \quad \int_{V} x_{i} x_{j} x_{k} \mathrm{~d} V=0 \quad \text { and } \int_{V} x_{i} x_{j} x_{k} x_{l} \mathrm{~d} V=\beta\left(\delta_{i j} \delta_{k l}+\delta_{i k} \delta_{j l}+\delta_{i l} \delta_{j k}\right)$$

  for some scalars $\alpha$ and $\beta$, which should be determined using suitable contractions of the indices or otherwise. Deduce the value of

  $$\int_{V} \mathbf{x} \times(\boldsymbol{\Omega} \times \mathbf{x}) \mathrm{d} V$$

  where $\boldsymbol{\Omega}$ is a constant vector.

  [You may assume that the most general isotropic tensor of rank 4 is

  $$\lambda \delta_{i j} \delta_{k l}+\mu \delta_{i k} \delta_{j l}+\nu \delta_{i l} \delta_{j k}$$

  where $\lambda, \mu$ and $\nu$ are scalars.]

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• Paper 3, Section II, C

  #Vector Calculus

  2011

  State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ bounded by a piecewise smooth surface $S$ with outward normal $\mathbf{n}$.

  Show, by suitable choice of $\mathbf{u}$, that

  $$\int_{V} \nabla f \mathrm{~d} V=\int_{S} f \mathrm{~d} \mathbf{S}$$

  for a scalar field $f(\mathbf{x})$.

  Let $V$ be the paraboloidal region given by $z \geqslant 0$ and $x^{2}+y^{2}+c z \leqslant a^{2}$, where $a$ and $c$ are positive constants. Verify that $(*)$ holds for the scalar field $f=x z$.

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• Paper 3, Section II, C

  #Vector Calculus

  2011

  The electric field $\mathbf{E}(\mathbf{x})$ due to a static charge distribution with density $\rho(\mathbf{x})$ satisfies

  $$\mathbf{E}=-\boldsymbol{\nabla} \phi, \quad \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\varepsilon_{0}},$$

  where $\phi(\mathbf{x})$ is the corresponding electrostatic potential and $\varepsilon_{0}$ is a constant.

  (a) Show that the total charge $Q$ contained within a closed surface $S$ is given by Gauss' Law

  $$Q=\varepsilon_{0} \int_{S} \mathbf{E} \cdot \mathrm{d} \mathbf{S} .$$

  Assuming spherical symmetry, deduce the electric field and potential due to a point charge $q$ at the origin i.e. for $\rho(\mathbf{x})=q \delta(\mathbf{x})$.

  (b) Let $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$, with potentials $\phi_{1}$ and $\phi_{2}$ respectively, be the solutions to (1) arising from two different charge distributions with densities $\rho_{1}$ and $\rho_{2}$. Show that

  $$\frac{1}{\varepsilon_{0}} \int_{V} \phi_{1} \rho_{2} \mathrm{~d} V+\int_{\partial V} \phi_{1} \nabla \phi_{2} \cdot \mathrm{d} \mathbf{S}=\frac{1}{\varepsilon_{0}} \int_{V} \phi_{2} \rho_{1} \mathrm{~d} V+\int_{\partial V} \phi_{2} \nabla \phi_{1} \cdot \mathrm{d} \mathbf{S}$$

  for any region $V$ with boundary $\partial V$, where $\mathrm{d} \mathbf{S}$ points out of $V$.

  (c) Suppose that $\rho_{1}(\mathbf{x})=0$ for $|\mathbf{x}| \leqslant a$ and that $\phi_{1}(\mathbf{x})=\Phi$, a constant, on $|\mathbf{x}|=a$. Use the results of (a) and (b) to show that

  $$\Phi=\frac{1}{4 \pi \varepsilon_{0}} \int_{r>a} \frac{\rho_{1}(\mathbf{x})}{r} \mathrm{~d} V$$

  [You may assume that $\phi_{1} \rightarrow 0$ as $|\mathbf{x}| \rightarrow \infty$ sufficiently rapidly that any integrals over the 'sphere at infinity' in (2) are zero.]

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• Paper 3, Section II, C

  #Vector Calculus

  2011

  The vector fields $\mathbf{A}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ obey the evolution equations

  $$\begin{aligned} \frac{\partial \mathbf{A}}{\partial t} &=\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{A})+\nabla \psi \\ \frac{\partial \mathbf{B}}{\partial t} &=(\mathbf{B} \cdot \nabla) \mathbf{u}-(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{B} \end{aligned}$$

  where $\mathbf{u}$ is a given vector field and $\psi$ is a given scalar field. Use suffix notation to show that the scalar field $h=\mathbf{A} \cdot \mathbf{B}$ obeys an evolution equation of the form

  $$\frac{\partial h}{\partial t}=\mathbf{B} \cdot \nabla f-\mathbf{u} \cdot \nabla h$$

  where the scalar field $f$ should be identified.

  Suppose that $\boldsymbol{\nabla} \cdot \mathbf{B}=0$ and $\boldsymbol{\nabla} \cdot \mathbf{u}=0$. Show that, if $\mathbf{u} \cdot \mathbf{n}=\mathbf{B} \cdot \mathbf{n}=0$ on the surface $S$ of a fixed volume $V$ with outward normal $\mathbf{n}$, then

  $$\frac{\mathrm{d} H}{\mathrm{~d} t}=0, \quad \text { where } H=\int_{V} h \mathrm{~d} V .$$

  Suppose that $\mathbf{A}=a r^{2} \sin \theta \mathbf{e}_{\theta}+r\left(a^{2}-r^{2}\right) \sin \theta \mathbf{e}_{\phi}$ with respect to spherical polar coordinates, and that $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. Show that

  $$h=a r^{2}\left(a^{2}+r^{2}\right) \sin ^{2} \theta$$

  and calculate the value of $H$ when $V$ is the sphere $r \leqslant a$.

  $$\left[\text { In spherical polar coordinates } \nabla \times \mathbf{F}=\frac{1}{r^{2} \sin \theta}\left|\begin{array}{ccc} \mathbf{e}_{r} & r \mathbf{e}_{\theta} & r \sin \theta \mathbf{e}_{\phi} \\ \partial / \partial r & \partial / \partial \theta & \partial / \partial \phi \\ F_{r} & r F_{\theta} & r \sin \theta F_{\phi} \end{array}\right|\right.$$

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• Paper 1, Section I, $1 \mathrm{C}$

  #Vectors and Matrices

  2011

  For $z, a \in \mathbb{C}$ define the principal value of $\log z$ and hence of $z^{a}$. Hence find all solutions to (i) $z^{\mathrm{i}}=1$ (ii) $z^{\mathrm{i}}+\bar{z}^{\mathrm{i}}=2 \mathrm{i}$,

  and sketch the curve $\left|z^{\mathrm{i}+1}\right|=1$.

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• Paper 1, Section I, A

  #Vectors and Matrices

  2011

  The matrix

  $$A=\left(\begin{array}{rr} 1 & -1 \\ 2 & 2 \\ -1 & 1 \end{array}\right)$$

  represents a linear map $\Phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ with respect to the bases

  $$B=\left\{\left(\begin{array}{l} 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \end{array}\right)\right\} \quad, \quad C=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)\right\}$$

  Find the matrix $A^{\prime}$ that represents $\Phi$ with respect to the bases

  $$B^{\prime}=\left\{\left(\begin{array}{l} 0 \\ 2 \end{array}\right),\left(\begin{array}{l} 2 \\ 0 \end{array}\right)\right\} \quad, \quad C^{\prime}=\left\{\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right)\right\}$$

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• Paper 1, Section II, $5 \mathrm{C}$

  #Vectors and Matrices

  2011

  Explain why each of the equations

  $$\begin{aligned} &\mathbf{x}=\mathbf{a}+\lambda \mathbf{b} \\ &\mathbf{x} \times \mathbf{c}=\mathbf{d} \end{aligned}$$

  describes a straight line, where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are constant vectors in $\mathbb{R}^{3}, \mathbf{b}$ and $\mathbf{c}$ are non-zero, $\mathbf{c} \cdot \mathbf{d}=0$ and $\lambda$ is a real parameter. Describe the geometrical relationship of a, $\mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ to the relevant line, assuming that $\mathbf{d} \neq \mathbf{0}$.

  Show that the solutions of (2) satisfy an equation of the form (1), defining $\mathbf{a}, \mathbf{b}$ and $\lambda(\mathbf{x})$ in terms of $\mathbf{c}$ and $\mathbf{d}$ such that $\mathbf{a} \cdot \mathbf{b}=0$ and $|\mathbf{b}|=|\mathbf{c}|$. Deduce that the conditions on $\mathbf{c}$ and $\mathbf{d}$ are sufficient for (2) to have solutions.

  For each of the lines described by (1) and (2), find the point $\mathbf{x}$ that is closest to a given fixed point $\mathbf{y}$.

  Find the line of intersection of the two planes $\mathbf{x} \cdot \mathbf{m}=\mu$ and $\mathbf{x} \cdot \mathbf{n}=\nu$, where $\mathbf{m}$ and $\mathbf{n}$ are constant unit vectors, $\mathbf{m} \times \mathbf{n} \neq \mathbf{0}$ and $\mu$ and $\nu$ are constants. Express your answer in each of the forms (1) and (2), giving both $\mathbf{a}$ and $\mathbf{d}$ as linear combinations of $\mathbf{m}$ and $\mathbf{n}$.

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• Paper 1, Section II, $6 \mathrm{~A}$

  #Vectors and Matrices

  2011

  The map $\Phi(\mathbf{x})=\mathbf{n} \times(\mathbf{x} \times \mathbf{n})+\alpha(\mathbf{n} \cdot \mathbf{x}) \mathbf{n}$ is defined for $\mathbf{x} \in \mathbb{R}^{3}$, where $\mathbf{n}$ is a unit vector in $\mathbb{R}^{3}$ and $\alpha$ is a constant.

  (a) Find the inverse map $\Phi^{-1}$, when it exists, and determine the values of $\alpha$ for which it does.

  (b) When $\Phi$ is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.

  (c) Let $\mathbf{y}=\Phi(\mathbf{x})$. Find the components $A_{i j}$ of the matrix $A$ such that $y_{i}=A_{i j} x_{j}$. When $\Phi$ is invertible, find the components of the matrix $B$ such that $x_{i}=B_{i j} y_{j}$.

  (d) Now let $A$ be as defined in (c) for the case $\mathbf{n}=\frac{1}{\sqrt{3}}(1,1,1)$, and let

  $$C=\frac{1}{3}\left(\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{array}\right)$$

  By analysing a suitable determinant, for all values of $\alpha$ find all vectors $\mathbf{x}$ such that $A \mathbf{x}=C \mathbf{x}$. Explain your results by interpreting $A$ and $C$ geometrically.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2011

  (a) Find the eigenvalues and eigenvectors of the matrix

  $$M=\left(\begin{array}{rrr} 2 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & -2 & 3 \end{array}\right)$$

  (b) Under what conditions on the $3 \times 3$ matrix $A$ and the vector $\mathbf{b}$ in $\mathbb{R}^{3}$ does the equation

  $$A \mathbf{x}=\mathbf{b}$$

  have 0,1 , or infinitely many solutions for the vector $\mathbf{x}$ in $\mathbb{R}^{3}$ ? Give clear, concise arguments to support your answer, explaining why just these three possibilities are allowed.

  (c) Using the results of $(\mathrm{a})$, or otherwise, find all solutions to $(*)$ when

  $$A=M-\lambda I \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{l} 4 \\ 3 \\ 2 \end{array}\right)$$

  in each of the cases $\lambda=0,1,2$.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2011

  (a) Let $M$ be a real symmetric $n \times n$ matrix. Prove the following.

  (i) Each eigenvalue of $M$ is real.

  (ii) Each eigenvector can be chosen to be real.

  (iii) Eigenvectors with different eigenvalues are orthogonal.

  (b) Let $A$ be a real antisymmetric $n \times n$ matrix. Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero.

  If $-\lambda^{2}$ and $-\mu^{2}$ are distinct, non-zero eigenvalues of $A^{2}$, show that there exist orthonormal vectors $\mathbf{u}, \mathbf{u}^{\prime}, \mathbf{w}, \mathbf{w}^{\prime}$ with

  $$\begin{array}{rlr} A \mathbf{u}=\lambda \mathbf{u}^{\prime}, & A \mathbf{w}=\mu \mathbf{w}^{\prime} \\ A \mathbf{u}^{\prime}=-\lambda \mathbf{u}, & A \mathbf{w}^{\prime}=-\mu \mathbf{w} \end{array}$$

  Part IA, 2011 List of Questions

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• , Complex Analysis

  #Analysis II

  2011

  Complex Analysis or Complex Methods

  Complex Methods

  Electromagnetism

  Fluid Dynamics

  Geometry

  Groups, Rings and Modules

  Linear Algebra

  Markov Chains

  Methods

  Metric and Topological Spaces

  Numerical Analysis

  Optimization

  Quantum Mechanics

  Statistics

  Variational Principles

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• Paper 2, Section I, E

  #Analysis II

  2011

  Define differentiability of a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Let $a>0$ be a constant. For which points $(x, y) \in \mathbb{R}^{2}$ is

  $$f(x, y)=|x|^{a}+|x-y|$$

  differentiable? Justify your answer.

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• Paper 3, Section I, $2 \mathrm{E}$

  #Analysis II

  2011

  Suppose $f$ is a uniformly continuous mapping from a metric space $X$ to a metric space $Y$. Prove that $f\left(x_{n}\right)$ is a Cauchy sequence in $Y$ for every Cauchy sequence $x_{n}$ in $X$.

  Let $f$ be a continuous mapping between metric spaces and suppose that $f$ has the property that $f\left(x_{n}\right)$ is a Cauchy sequence whenever $x_{n}$ is a Cauchy sequence. Is it true that $f$ must be uniformly continuous? Justify your answer.

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• Paper 4, Section I, E

  #Analysis II

  2011

  Let $B[0,1]$ denote the set of bounded real-valued functions on $[0,1]$. A distance $d$ on $B[0,1]$ is defined by

  $$d(f, g)=\sup _{x \in[0,1]}|f(x)-g(x)| .$$

  Given that $(B[0,1], d)$ is a metric space, show that it is complete. Show that the subset $C[0,1] \subset B[0,1]$ of continuous functions is a closed set.

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• Paper 1, Section II, E

  #Analysis II

  2011

  What is meant by saying that a sequence of functions $f_{n}$ converges uniformly to a function $f$ ?

  Let $f_{n}$ be a sequence of differentiable functions on $[a, b]$ with $f_{n}^{\prime}$ continuous and such that $f_{n}\left(x_{0}\right)$ converges for some point $x_{0} \in[a, b]$. Assume in addition that $f_{n}^{\prime}$ converges uniformly on $[a, b]$. Prove that $f_{n}$ converges uniformly to a differentiable function $f$ on $[a, b]$ and $f^{\prime}(x)=\lim _{n \rightarrow \infty} f_{n}^{\prime}(x)$ for all $x \in[a, b]$. [You may assume that the uniform limit of continuous functions is continuous.]

  Show that the series

  $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^{s}}$$

  converges for $s>1$ and is uniformly convergent on $[1+\varepsilon, \infty)$ for any $\varepsilon>0$. Show that $\zeta(s)$ is differentiable on $(1, \infty)$ and

  $$\zeta^{\prime}(s)=-\sum_{n=2}^{\infty} \frac{\log n}{n^{s}}$$

  [You may use the Weierstrass $M$-test provided it is clearly stated.]

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• Paper 2, Section II, E

  #Analysis II

  2011

  What is meant by saying that two norms on a real vector space are Lipschitz equivalent?

  Show that any two norms on $\mathbb{R}^{n}$ are Lipschitz equivalent. [You may assume that a continuous function on a closed bounded set in $\mathbb{R}^{n}$ has closed bounded image.]

  Show that $\|f\|_{1}=\int_{-1}^{1}|f(x)| d x$ defines a norm on the space $C[-1,1]$ of continuous real-valued functions on $[-1,1]$. Is it Lipschitz equivalent to the uniform norm? Justify your answer. Prove that the normed space $\left(C[-1,1],\|\cdot\|_{1}\right)$ is not complete.

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• Paper 3, Section II, E

  #Analysis II

  2011

  Consider a map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$.

  Assume $f$ is differentiable at $x$ and let $D_{x} f$ denote the derivative of $f$ at $x$. Show that

  $$D_{x} f(v)=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$$

  for any $v \in \mathbb{R}^{n}$.

  Assume now that $f$ is such that for some fixed $x$ and for every $v \in \mathbb{R}^{n}$ the limit

  $$\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$$

  exists. Is it true that $f$ is differentiable at $x ?$ Justify your answer.

  Let $M_{k}$ denote the set of all $k \times k$ real matrices which is identified with $\mathbb{R}^{k^{2}}$. Consider the function $f: M_{k} \rightarrow M_{k}$ given by $f(A)=A^{3}$. Explain why $f$ is differentiable. Show that the derivative of $f$ at the matrix $A$ is given by

  $$D_{A} f(H)=H A^{2}+A H A+A^{2} H$$

  for any matrix $H \in M_{k}$. State carefully the inverse function theorem and use it to prove that there exist open sets $U$ and $V$ containing the identity matrix such that given $B \in V$ there exists a unique $A \in U$ such that $A^{3}=B$.

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• Paper 4, Section II, E

  #Analysis II

  2011

  Define a contraction mapping and state the contraction mapping theorem.

  Let $(X, d)$ be a non-empty complete metric space and let $\phi: X \rightarrow X$ be a map. Set $\phi^{1}=\phi$ and $\phi^{n+1}=\phi \circ \phi^{n}$. Assume that for some integer $r \geqslant 1, \phi^{r}$ is a contraction mapping. Show that $\phi$ has a unique fixed point $y$ and that any $x \in X$ has the property that $\phi^{n}(x) \rightarrow y$ as $n \rightarrow \infty$.

  Let $C[0,1]$ be the set of continuous real-valued functions on $[0,1]$ with the uniform norm. Suppose $T: C[0,1] \rightarrow C[0,1]$ is defined by

  $$T(f)(x)=\int_{0}^{x} f(t) d t$$

  for all $x \in[0,1]$ and $f \in C[0,1]$. Show that $T$ is not a contraction mapping but that $T^{2}$ is.

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• Paper 4, Section I, E

  #Complex Analysis

  2011

  Let $f(z)$ be an analytic function in an open subset $U$ of the complex plane. Prove that $f$ has derivatives of all orders at any point $z$ in $U$. [You may assume Cauchy's integral formula provided it is clearly stated.]

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• Paper 3, Section II, E

  #Complex Analysis

  2011

  Let $g: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function such that

  $$\int_{\Gamma} g(z) d z=0$$

  for any closed curve $\Gamma$ which is the boundary of a rectangle in $\mathbb{C}$ with sides parallel to the real and imaginary axes. Prove that $g$ is analytic.

  Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be continuous. Suppose in addition that $f$ is analytic at every point $z \in \mathbb{C}$ with non-zero imaginary part. Show that $f$ is analytic at every point in $\mathbb{C} .$

  Let $\mathbb{H}$ be the upper half-plane of complex numbers $z$ with positive imaginary part $\Im(z)>0$. Consider a continuous function $F: \mathbb{H} \cup \mathbb{R} \rightarrow \mathbb{C}$ such that $F$ is analytic on $\mathbb{H}$ and $F(\mathbb{R}) \subset \mathbb{R}$. Define $f: \mathbb{C} \rightarrow \mathbb{C}$ by

  $$f(z)= \begin{cases}F(z) & \text { if } \Im(z) \geqslant 0 \\ \overline{F(\bar{z})} & \text { if } \Im(z) \leqslant 0\end{cases}$$

  Show that $f$ is analytic.

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• Paper 1, Section I, A

  #Complex Analysis or Complex Methods

  2011

  Derive the Cauchy-Riemann equations satisfied by the real and imaginary parts of a complex analytic function $f(z)$.

  If $|f(z)|$ is constant on $|z|<1$, prove that $f(z)$ is constant on $|z|<1$.

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• Paper 1, Section II, A

  #Complex Analysis or Complex Methods

  2011

  (i) Let $-1<\alpha<0$ and let

  $$\begin{aligned} &f(z)=\frac{\log (z-\alpha)}{z} \text { where }-\pi \leqslant \arg (z-\alpha)<\pi \\ &g(z)=\frac{\log z}{z} \quad \text { where }-\pi \leqslant \arg (z)<\pi \end{aligned}$$

  Here the logarithms take their principal values. Give a sketch to indicate the positions of the branch cuts implied by the definitions of $f(z)$ and $g(z)$.

  (ii) Let $h(z)=f(z)-g(z)$. Explain why $h(z)$ is analytic in the annulus $1 \leqslant|z| \leqslant R$ for any $R>1$. Obtain the first three terms of the Laurent expansion for $h(z)$ around $z=0$ in this annulus and hence evaluate

  $$\oint_{|z|=2} h(z) d z$$

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• Paper 2, Section II, A

  #Complex Analysis or Complex Methods

  2011

  (i) Let $C$ be an anticlockwise contour defined by a square with vertices at $z=x+i y$ where

  $$|x|=|y|=\left(2 N+\frac{1}{2}\right) \pi$$

  for large integer $N$. Let

  $$I=\oint_{C} \frac{\pi \cot z}{(z+\pi a)^{4}} d z$$

  Assuming that $I \rightarrow 0$ as $N \rightarrow \infty$, prove that, if $a$ is not an integer, then

  $$\sum_{n=-\infty}^{\infty} \frac{1}{(n+a)^{4}}=\frac{\pi^{4}}{3 \sin ^{2}(\pi a)}\left(\frac{3}{\sin ^{2}(\pi a)}-2\right) .$$

  (ii) Deduce the value of

  $$\sum_{n=-\infty}^{\infty} \frac{1}{\left(n+\frac{1}{2}\right)^{4}}$$

  (iii) Briefly justify the assumption that $I \rightarrow 0$ as $N \rightarrow \infty$.

  [Hint: For part (iii) it is sufficient to consider, at most, one vertical side of the square and one horizontal side and to use a symmetry argument for the remaining sides.]

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• Paper 3, Section I, D

  #Complex Methods

  2011

  Write down the function $\psi(u, v)$ that satisfies

  $$\frac{\partial^{2} \psi}{\partial u^{2}}+\frac{\partial^{2} \psi}{\partial v^{2}}=0, \quad \psi\left(-\frac{1}{2}, v\right)=-1, \quad \psi\left(\frac{1}{2}, v\right)=1$$

  The circular arcs $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ in the complex $z$-plane are defined by

  $$|z+1|=1, z \neq 0 \text { and }|z-1|=1, z \neq 0,$$

  respectively. You may assume without proof that the mapping from the complex $z$-plane to the complex $\zeta$-plane defined by

  $$\zeta=\frac{1}{z}$$

  takes $\mathcal{C}_{1}$ to the line $u=-\frac{1}{2}$ and $\mathcal{C}_{2}$ to the line $u=\frac{1}{2}$, where $\zeta=u+i v$, and that the region $\mathcal{D}$ in the $z$-plane exterior to both the circles $|z+1|=1$ and $|z-1|=1$ maps to the region in the $\zeta$-plane given by $-\frac{1}{2}<u<\frac{1}{2}$.

  Use the above mapping to solve the problem

  $$\nabla^{2} \phi=0 \quad \text { in } \mathcal{D}, \quad \phi=-1 \text { on } \mathcal{C}_{1} \text { and } \phi=1 \text { on } \mathcal{C}_{2}$$

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• Paper 4, Section II, D

  #Complex Methods

  2011

  State and prove the convolution theorem for Laplace transforms.

  Use Laplace transforms to solve

  $$2 f^{\prime}(t)-\int_{0}^{t}(t-\tau)^{2} f(\tau) d \tau=4 t H(t)$$

  with $f(0)=0$, where $H(t)$ is the Heaviside function. You may assume that the Laplace transform, $\widehat{f}(s)$, of $f(t)$ exists for Re $s$ sufficiently large.

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• Paper 2, Section I, $\mathbf{6 C}$

  #Electromagnetism

  2011

  Maxwell's equations are

  $$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0, \quad \nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t} \end{gathered}$$

  Find the equation relating $\rho$ and $\mathbf{J}$ that must be satisfied for consistency, and give the interpretation of this equation.

  Now consider the "magnetic limit" where $\rho=0$ and the term $\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t}$ is neglected. Let $\mathbf{A}$ be a vector potential satisfying the gauge condition $\boldsymbol{\nabla} \cdot \mathbf{A}=0$, and assume the scalar potential vanishes. Find expressions for $\mathbf{E}$ and $\mathbf{B}$ in terms of $\mathbf{A}$ and show that Maxwell's equations are all satisfied provided $\mathbf{A}$ satisfies the appropriate Poisson equation.

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• Paper 4, Section I, C

  #Electromagnetism

  2011

  A plane electromagnetic wave in a vacuum has electric field

  $$\mathbf{E}=\left(E_{0} \sin k(z-c t), 0,0\right)$$

  What are the wavevector, polarization vector and speed of the wave? Using Maxwell's equations, find the magnetic field B. Assuming the scalar potential vanishes, find a possible vector potential $\mathbf{A}$ for this wave, and verify that it gives the correct $\mathbf{E}$ and $\mathbf{B}$.

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• Paper 1, Section II, D

  #Electromagnetism

  2011

  Starting from the relevant Maxwell equation, derive Gauss's law in integral form.

  Use Gauss's law to obtain the potential at a distance $r$ from an infinite straight wire with charge $\lambda$ per unit length.

  Write down the potential due to two infinite wires parallel to the $z$-axis, one at $x=y=0$ with charge $\lambda$ per unit length and the other at $x=0, y=d$ with charge $-\lambda$ per unit length.

  Find the potential and the electric field in the limit $d \rightarrow 0$ with $\lambda d=p$ where $p$ is fixed. Sketch the equipotentials and the electric field lines.

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• Paper 2, Section II, C

  #Electromagnetism

  2011

  (i) Consider an infinitely long solenoid parallel to the $z$-axis whose cross section is a simple closed curve of arbitrary shape. A current $I$, per unit length of the solenoid, flows around the solenoid parallel to the $x-y$ plane. Show using the relevant Maxwell equation that the magnetic field $\mathbf{B}$ inside the solenoid is uniform, and calculate its magnitude.

  (ii) A wire loop in the shape of a regular hexagon of side length $a$ carries a current $I$. Use the Biot-Savart law to calculate $\mathbf{B}$ at the centre of the loop.

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• Paper 3, Section II, C

  #Electromagnetism

  2011

  Show, using the vacuum Maxwell equations, that for any volume $V$ with surface $S$,

  $$\frac{d}{d t} \int_{V}\left(\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}\right) d V=\int_{S}\left(-\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}\right) \cdot \mathbf{d} \mathbf{S}$$

  What is the interpretation of this equation?

  A uniform straight wire, with a circular cross section of radius $r$, has conductivity $\sigma$ and carries a current $I$. Calculate $\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$ at the surface of the wire, and hence find the flux of $\frac{1}{\mu_{0}} \mathbf{E} \times \mathbf{B}$ into unit length of the wire. Relate your result to the resistance of the wire, and the rate of energy dissipation.

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• Paper 1, Section I, B

  #Fluid Dynamics

  2011

  Inviscid fluid is contained in a square vessel with sides of length $\pi L$ lying between $x=0, \pi L, y=0, \pi L$. The base of the container is at $z=-H$ where $H \gg L$ and the horizontal surface is at $z=0$ when the fluid is at rest. The variation of pressure of the air above the fluid may be neglected.

  Small amplitude surface waves are excited in the vessel.

  (i) Now let $H \rightarrow \infty$. Explain why on dimensional grounds the frequencies $\omega$ of such waves are of the form

  $$\omega=\left(\frac{\gamma g}{L}\right)^{\frac{1}{2}}$$

  for some positive dimensionless constants $\gamma$, where $g$ is the gravitational acceleration.

  It is given that the velocity potential $\phi$ is of the form

  $$\phi(x, y, z) \approx C \cos (m x / L) \cos (n y / L) \mathrm{e}^{\gamma z / L}$$

  where $m$ and $n$ are integers and $C$ is a constant.

  (ii) Why do cosines, rather than sines, appear in this expression?

  (iii) Give an expression for $\gamma$ in terms of $m$ and $n$.

  (iv) Give all possible values that $\gamma^{2}$ can take between 1 and 10 inclusive. How many different solutions for $\phi$ correspond to each of these values of $\gamma^{2} ?$

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• Paper 2, Section I, D

  #Fluid Dynamics

  2011

  A body of volume $V$ lies totally submerged in a motionless fluid of uniform density $\rho$. Show that the force $\mathbf{F}$ on the body is given by

  $$\mathbf{F}=-\int_{S}\left(p-p_{0}\right) \mathbf{n} d S$$

  where $p$ is the pressure in the fluid and $p_{0}$ is atmospheric pressure. You may use without proof the generalised divergence theorem in the form

  $$\int_{S} \phi \mathbf{n} d S=\int_{V} \boldsymbol{\nabla} \phi d V$$

  Deduce that

  $$\mathbf{F}=\rho g V \hat{\mathbf{z}},$$

  where $\hat{\mathbf{z}}$ is the vertically upward unit vector. Interpret this result.

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• Paper 1, Section II, B

  #Fluid Dynamics

  2011

  A spherical bubble in an incompressible fluid of density $\rho$ has radius $a(t)$. Write down an expression for the velocity field at a radius $R \geqslant a$.

  The pressure far from the bubble is $p_{\infty}$. What is the pressure at radius $R$ ?

  Find conditions on $a$ and its time derivatives that ensure that the maximum pressure in the fluid is reached at a radius $R_{\max }$ where $a<R_{\max }<\infty$. Give an expression for this maximum pressure when the conditions hold.

  Give the most general form of $a(t)$ that ensures that the pressure at $R=a(t)$ is $p_{\infty}$ for all time.

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• Paper 3, Section II, $18 \mathrm{D}$

  #Fluid Dynamics

  2011

  Water of constant density $\rho$ flows steadily through a long cylindrical tube, the wall of which is elastic. The exterior radius of the tube at a distance $z$ along the tube, $r(z)$, is determined by the pressure in the tube, $p(z)$, according to

  $$r(z)=r_{0}+b\left(p(z)-p_{0}\right)$$

  where $r_{0}$ and $p_{0}$ are the radius and pressure far upstream $(z \rightarrow-\infty)$, and $b$ is a positive constant.

  The interior radius of the tube is $r(z)-h(z)$, where $h(z)$, the thickness of the wall, is a given slowly varying function of $z$ which is zero at both ends of the pipe. The velocity of the water in the pipe is $u(z)$ and the water enters the pipe at velocity $V$.

  Show that $u(z)$ satisfies

  $$H=1-v^{-\frac{1}{2}}+\frac{1}{4} k\left(1-v^{2}\right)$$

  where $H=\frac{h}{r_{0}}, v=\frac{u}{V}$ and $k=\frac{2 b \rho V^{2}}{r_{0}}$. Sketch the graph of $H$ against $v$.

  Let $H_{m}$ be the maximum value of $H$ in the tube. Show that the flow is only possible if $H_{m}$ does not exceed a certain critical value $H_{c}$. Find $H_{c}$ in terms of $k$.

  Show that, under conditions to be determined (which include a condition on the value of $k)$, the water can leave the pipe with speed less than $V$.

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• Paper 4, Section II, D

  #Fluid Dynamics

  2011

  Show that an irrotational incompressible flow can be determined from a velocity potential $\phi$ that satisfies $\nabla^{2} \phi=0$.

  Given that the general solution of $\nabla^{2} \phi=0$ in plane polar coordinates is

  $$\phi=\sum_{n=-\infty}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right) r^{n}+c \log r+b \theta$$

  obtain the corresponding fluid velocity.

  A two-dimensional irrotational incompressible fluid flows past the circular disc with boundary $r=a$. For large $r$, the flow is uniform and parallel to the $x$-axis $(x=r \cos \theta)$. Write down the boundary conditions for large $r$ and on $r=a$, and hence derive the velocity potential in the form

  $$\phi=U\left(r+\frac{a^{2}}{r}\right) \cos \theta+\frac{\kappa \theta}{2 \pi}$$

  where $\kappa$ is the circulation.

  Show that the acceleration of the fluid at $r=a$ and $\theta=0$ is

  $$\frac{\kappa}{2 \pi a^{2}}\left(-\frac{\kappa}{2 \pi a} \mathbf{e}_{r}-2 U \mathbf{e}_{\theta}\right)$$

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• Paper 1, Section I, F

  #Geometry

  2011

  Suppose that $H \subseteq \mathbb{C}$ is the upper half-plane, $H=\{x+i y \mid x, y \in \mathbb{R}, y>0\}$. Using the Riemannian metric $d s^{2}=\frac{d x^{2}+d y^{2}}{y^{2}}$, define the length of a curve $\gamma$ and the area of a region $\Omega$ in $H$.

  Find the area of

  $$\Omega=\left\{x+i y|| x \mid \leqslant \frac{1}{2}, x^{2}+y^{2} \geqslant 1\right\}$$

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• Paper 3, Section I, F

  #Geometry

  2011

  Let $R(x, \theta)$ denote anti-clockwise rotation of the Euclidean plane $\mathbb{R}^{2}$ through an angle $\theta$ about a point $x$.

  Show that $R(x, \theta)$ is a composite of two reflexions.

  Assume $\theta, \phi \in(0, \pi)$. Show that the composite $R(y, \phi) \cdot R(x, \theta)$ is also a rotation $R(z, \psi)$. Find $z$ and $\psi$.

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• Paper 2, Section II, F

  #Geometry

  2011

  Suppose that $\pi: S^{2} \rightarrow \mathbb{C}_{\infty}$ is stereographic projection. Show that, via $\pi$, every rotation of $S^{2}$ corresponds to a Möbius transformation in $P S U(2)$.

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• Paper 3, Section II, F

  #Geometry

  2011

  Suppose that $\eta(u)=(f(u), 0, g(u))$ is a unit speed curve in $\mathbb{R}^{3}$. Show that the corresponding surface of revolution $S$ obtained by rotating this curve about the $z$-axis has Gaussian curvature $K=-\left(d^{2} f / d u^{2}\right) / f$.

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• Paper 4, Section II, F

  #Geometry

  2011

  Suppose that $P$ is a point on a Riemannian surface $S$. Explain the notion of geodesic polar co-ordinates on $S$ in a neighbourhood of $P$, and prove that if $C$ is a geodesic circle centred at $P$ of small positive radius, then the geodesics through $P$ meet $C$ at right angles.

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• Paper 2, Section I, F

  #Groups, Rings and Modules

  2011

  Show that the quaternion group $Q_{8}=\{\pm 1, \pm i, \pm j, \pm k\}$, with $i j=k=-j i$, $i^{2}=j^{2}=k^{2}=-1$, is not isomorphic to the symmetry group $D_{8}$ of the square.

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• Paper 3, Section I, $1 F$

  #Groups, Rings and Modules

  2011

  Suppose that $A$ is an integral domain containing a field $K$ and that $A$ is finitedimensional as a $K$-vector space. Prove that $A$ is a field.

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• Paper 4, Section I, F

  #Groups, Rings and Modules

  2011

  A ring $R$ satisfies the descending chain condition (DCC) on ideals if, for every sequence $I_{1} \supseteq I_{2} \supseteq I_{3} \supseteq \ldots$ of ideals in $R$, there exists $n$ with $I_{n}=I_{n+1}=I_{n+2}=\ldots$ Show that $\mathbb{Z}$ does not satisfy the DCC on ideals.

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• Paper 1, Section II, F

  #Groups, Rings and Modules

  2011

  (i) Suppose that $G$ is a finite group of order $p^{n} r$, where $p$ is prime and does not divide $r$. Prove the first Sylow theorem, that $G$ has at least one subgroup of order $p^{n}$, and state the remaining Sylow theorems without proof.

  (ii) Suppose that $p, q$ are distinct primes. Show that there is no simple group of order $p q$.

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• Paper 2, Section II, F

  #Groups, Rings and Modules

  2011

  Define the notion of a Euclidean domain and show that $\mathbb{Z}[i]$ is Euclidean.

  Is $4+i$ prime in $\mathbb{Z}[i]$ ?

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• Paper 3, Section II, F

  #Groups, Rings and Modules

  2011

  Suppose that $A$ is a matrix over $\mathbb{Z}$. What does it mean to say that $A$ can be brought to Smith normal form?

  Show that the structure theorem for finitely generated modules over $\mathbb{Z}$ (which you should state) follows from the existence of Smith normal forms for matrices over $\mathbb{Z}$.

  Bring the matrix $\left(\begin{array}{cc}-4 & -6 \\ 2 & 2\end{array}\right)$ to Smith normal form.

  Suppose that $M$ is the $\mathbb{Z}$-module with generators $e_{1}, e_{2}$, subject to the relations

  $$-4 e_{1}+2 e_{2}=-6 e_{1}+2 e_{2}=0$$

  Describe $M$ in terms of the structure theorem.

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• Paper 4, Section II, F

  #Groups, Rings and Modules

  2011

  State and prove the Hilbert Basis Theorem.

  Is every ring Noetherian? Justify your answer.

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• Paper 1, Section I, G

  #Linear Algebra

  2011

  (i) State the rank-nullity theorem for a linear map between finite-dimensional vector spaces.

  (ii) Show that a linear transformation $f: V \rightarrow V$ of a finite-dimensional vector space $V$ is bijective if it is injective or surjective.

  (iii) Let $V$ be the $\mathbb{R}$-vector space $\mathbb{R}[X]$ of all polynomials in $X$ with coefficients in $\mathbb{R}$. Give an example of a linear transformation $f: V \rightarrow V$ which is surjective but not bijective.

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• Paper 2, Section $I$, G

  #Linear Algebra

  2011

  Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space with an inner product. Let $W$ be an $m$-dimensional subspace of $V$ and $W^{\perp}$ its orthogonal complement, so that every element $v \in V$ can be uniquely written as $v=w+w^{\prime}$ for $w \in W$ and $w^{\prime} \in W^{\perp}$.

  The reflection map with respect to $W$ is defined as the linear map

  $$f_{W}: V \ni w+w^{\prime} \longmapsto w-w^{\prime} \in V$$

  Show that $f_{W}$ is an orthogonal transformation with respect to the inner product, and find its determinant.

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• Paper 4, Section I, G

  #Linear Algebra

  2011

  (i) Let $V$ be a vector space over a field $F$, and $W_{1}, W_{2}$ subspaces of $V$. Define the subset $W_{1}+W_{2}$ of $V$, and show that $W_{1}+W_{2}$ and $W_{1} \cap W_{2}$ are subspaces of $V$.

  (ii) When $W_{1}, W_{2}$ are finite-dimensional, state a formula for $\operatorname{dim}\left(W_{1}+W_{2}\right)$ in terms of $\operatorname{dim} W_{1}, \operatorname{dim} W_{2}$ and $\operatorname{dim}\left(W_{1} \cap W_{2}\right)$.

  (iii) Let $V$ be the $\mathbb{R}$-vector space of all $n \times n$ matrices over $\mathbb{R}$. Let $S$ be the subspace of all symmetric matrices and $T$ the subspace of all upper triangular matrices (the matrices $\left(a_{i j}\right)$ such that $a_{i j}=0$ whenever $\left.i>j\right)$. Find $\operatorname{dim} S, \operatorname{dim} T, \operatorname{dim}(S \cap T)$ and $\operatorname{dim}(S+T)$. Briefly justify your answer.

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• Paper 1, Section II, G

  #Linear Algebra

  2011

  Let $V, W$ be finite-dimensional vector spaces over a field $F$ and $f: V \rightarrow W$ a linear map.

  (i) Show that $f$ is injective if and only if the image of every linearly independent subset of $V$ is linearly independent in $W$.

  (ii) Define the dual space $V^{*}$ of $V$ and the dual map $f^{*}: W^{*} \rightarrow V^{*}$.

  (iii) Show that $f$ is surjective if and only if the image under $f^{*}$ of every linearly independent subset of $W^{*}$ is linearly independent in $V^{*}$.

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• Paper 2, Section II, G

  #Linear Algebra

  2011

  Let $n$ be a positive integer, and let $V$ be a $\mathbb{C}$-vector space of complex-valued functions on $\mathbb{R}$, generated by the set $\{\cos k x, \sin k x ; k=0,1, \ldots, n-1\}$.

  (i) Let $\langle f, g\rangle=\int_{0}^{2 \pi} f(x) \overline{g(x)} d x$ for $f, g \in V$. Show that this is a positive definite Hermitian form on $V$.

  (ii) Let $\Delta(f)=\frac{d^{2}}{d x^{2}} f(x)$. Show that $\Delta$ is a self-adjoint linear transformation of $V$ with respect to the form defined in (i).

  (iii) Find an orthonormal basis of $V$ with respect to the form defined in (i), which consists of eigenvectors of $\Delta$.

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• Paper 3, Section II, G

  #Linear Algebra

  2011

  (i) Let $A$ be an $n \times n$ complex matrix and $f(X)$ a polynomial with complex coefficients. By considering the Jordan normal form of $A$ or otherwise, show that if the eigenvalues of $A$ are $\lambda_{1}, \ldots, \lambda_{n}$ then the eigenvalues of $f(A)$ are $f\left(\lambda_{1}\right), \ldots, f\left(\lambda_{n}\right)$.

  (ii) Let $B=\left(\begin{array}{llll}a & d & c & b \\ b & a & d & c \\ c & b & a & d \\ d & c & b & a\end{array}\right)$. Write $B$ as $B=f(A)$ for a polynomial $f$ with $A=\left(\begin{array}{llll}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array}\right)$, and find the eigenvalues of $B$

  [Hint: compute the powers of $A$.]

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• Paper 4, Section II, G

  #Linear Algebra

  2011

  Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space and $f, g: V \rightarrow V$ linear transformations. Suppose $f$ is invertible and diagonalisable, and $f \circ g=t \cdot(g \circ f)$ for some real number $t>1$.

  (i) Show that $g$ is nilpotent, i.e. some positive power of $g$ is 0 .

  (ii) Suppose that there is a non-zero vector $v \in V$ with $f(v)=v$ and $g^{n-1}(v) \neq 0$. Determine the diagonal form of $f$.

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• Paper 3, Section I, H

  #Markov Chains

  2011

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain with state space $S$.

  (i) What does it mean to say that $\left(X_{n}\right)_{n} \geqslant 0$ has the strong Markov property? Your answer should include the definition of the term stopping time.

  (ii) Show that

  $$\mathbb{P}\left(X_{n}=i \text { at least } k \text { times } \mid X_{0}=i\right)=\left[\mathbb{P}\left(X_{n}=i \text { at least once } \mid X_{0}=i\right)\right]^{k}$$

  for a state $i \in S$. You may use without proof the fact that $\left(X_{n}\right)_{n \geqslant 0}$ has the strong Markov property.

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• Paper 4, Section I, H

  #Markov Chains

  2011

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain on a state space $S$, and let $p_{i j}(n)=\mathbb{P}\left(X_{n}=j \mid X_{0}=i\right)$.

  (i) What does the term communicating class mean in terms of this chain?

  (ii) Show that $p_{i i}(m+n) \geqslant p_{i j}(m) p_{j i}(n)$.

  (iii) The period $d_{i}$ of a state $i$ is defined to be

  $$d_{i}=\operatorname{gcd}\left\{n \geqslant 1: p_{i i}(n)>0\right\}$$

  Show that if $i$ and $j$ are in the same communicating class and $p_{j j}(r)>0$, then $d_{i}$ divides $r$.

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• Paper 1, Section II, H

  #Markov Chains

  2011

  Let $P=\left(p_{i j}\right)_{i, j \in S}$ be the transition matrix for an irreducible Markov chain on the finite state space $S$.

  (i) What does it mean to say $\pi$ is the invariant distribution for the chain?

  (ii) What does it mean to say the chain is in detailed balance with respect to $\pi$ ?

  (iii) A symmetric random walk on a connected finite graph is the Markov chain whose state space is the set of vertices of the graph and whose transition probabilities are

  $$p_{i j}= \begin{cases}1 / D_{i} & \text { if } j \text { is adjacent to } i \\ 0 & \text { otherwise }\end{cases}$$

  where $D_{i}$ is the number of vertices adjacent to vertex $i$. Show that the random walk is in detailed balance with respect to its invariant distribution.

  (iv) Let $\pi$ be the invariant distribution for the transition matrix $P$, and define an inner product for vectors $x, y \in \mathbb{R}^{S}$ by the formula

  $$\langle x, y\rangle=\sum_{i \in S} x_{i} \pi_{i} y_{i}$$

  Show that the equation

  $$\langle x, P y\rangle=\langle P x, y\rangle$$

  holds for all vectors $x, y \in \mathbb{R}^{S}$ if and only if the chain is in detailed balance with respect to $\pi$. [Here $z \in \mathbb{R}^{S}$ means $z=\left(z_{i}\right)_{i \in S}$.]

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• Paper 2, Section II, H

  #Markov Chains

  2011

  (i) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain on the finite state space $S$ with transition matrix $P$. Fix a subset $A \subseteq S$, and let

  $$H=\inf \left\{n \geqslant 0: X_{n} \in A\right\} .$$

  Fix a function $g$ on $S$ such that $0<g(i) \leqslant 1$ for all $i \in S$, and let

  $$V_{i}=\mathbb{E}\left[\prod_{n=0}^{H-1} g\left(X_{n}\right) \mid X_{0}=i\right]$$

  where $\prod_{n=0}^{-1} a_{n}=1$ by convention. Show that

  $$V_{i}= \begin{cases}1 & \text { if } i \in A \\ g(i) \sum_{j \in S} P_{i j} V_{j} & \text { otherwise. }\end{cases}$$

  (ii) A flea lives on a polyhedron with $N$ vertices, labelled $1, \ldots, N$. It hops from vertex to vertex in the following manner: if one day it is on vertex $i>1$, the next day it hops to one of the vertices labelled $1, \ldots, i-1$ with equal probability, and it dies upon reaching vertex 1. Let $X_{n}$ be the position of the flea on day $n$. What are the transition probabilities for the Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ ?

  (iii) Let $H$ be the number of days the flea is alive, and let

  $$V_{i}=\mathbb{E}\left(s^{H} \mid X_{0}=i\right)$$

  where $s$ is a real number such that $0<s \leqslant 1$. Show that $V_{1}=1$ and

  $$\frac{i}{s} V_{i+1}=V_{i}+\frac{i-1}{s} V_{i}$$

  for $i \geqslant 1$. Conclude that

  $$\mathbb{E}\left(s^{H} \mid X_{0}=N\right)=\prod_{i=1}^{N-1}\left(1+\frac{s-1}{i}\right)$$

  [Hint. Use part (i) with $A=\{1\}$ and a well-chosen function $g$. ]

  (iv) Show that

  $$\mathbb{E}\left(H \mid X_{0}=N\right)=\sum_{i=1}^{N-1} \frac{1}{i}$$

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• Paper 2, Section I, A

  #Methods

  2011

  The Legendre equation is

  $$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+n(n+1) y=0$$

  for $-1 \leqslant x \leqslant 1$ and non-negative integers $n$.

  Write the Legendre equation as an eigenvalue equation for an operator $L$ in SturmLiouville form. Show that $L$ is self-adjoint and find the orthogonality relation between the eigenfunctions.

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• Paper 3, Section I, A

  #Methods

  2011

  The Fourier transform $\tilde{h}(k)$ of the function $h(x)$ is defined by

  $$\tilde{h}(k)=\int_{-\infty}^{\infty} h(x) e^{-i k x} d x$$

  (i) State the inverse Fourier transform formula expressing $h(x)$ in terms of $\widetilde{h}(k)$.

  (ii) State the convolution theorem for Fourier transforms.

  (iii) Find the Fourier transform of the function $f(x)=e^{-|x|}$. Hence show that the convolution of the function $f(x)=e^{-|x|}$ with itself is given by the integral expression

  $$\frac{2}{\pi} \int_{-\infty}^{\infty} \frac{e^{i k x}}{\left(1+k^{2}\right)^{2}} d k$$

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• Paper 4, Section I, A

  #Methods

  2011

  Use the method of characteristics to find a continuous solution $u(x, y)$ of the equation

  $$y \frac{\partial u}{\partial x}+x \frac{\partial u}{\partial y}=0$$

  subject to the condition $u(0, y)=y^{4}$.

  In which region of the plane is the solution uniquely determined?

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• Paper 1, Section II, A

  #Methods

  2011

  Let $f(t)$ be a real function defined on an interval $(-T, T)$ with Fourier series

  $$f(t)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n} \cos \frac{n \pi t}{T}+b_{n} \sin \frac{n \pi t}{T}\right)$$

  State and prove Parseval's theorem for $f(t)$ and its Fourier series. Write down the formulae for $a_{0}, a_{n}$ and $b_{n}$ in terms of $f(t), \cos \frac{n \pi t}{T}$ and $\sin \frac{n \pi t}{T}$.

  Find the Fourier series of the square wave function defined on $(-\pi, \pi)$ by

  $$g(t)=\left\{\begin{array}{lr} 0 & -\pi<t \leqslant 0 \\ 1 & 0<t<\pi \end{array}\right.$$

  Hence evaluate

  $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2 k+1)}$$

  Using some of the above results evaluate

  $$\sum_{k=0}^{\infty} \frac{1}{(2 k+1)^{2}}$$

  What is the sum of the Fourier series for $g(t)$ at $t=0$ ? Comment on your answer.

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• Paper 2, Section II, A

  #Methods

  2011

  Use a Green's function to find an integral expression for the solution of the equation

  $$\frac{d^{2} \theta}{d t^{2}}+4 \frac{d \theta}{d t}+29 \theta=f(t)$$

  for $t \geqslant 0$ subject to the initial conditions

  $$\theta(0)=0 \quad \text { and } \quad \frac{d \theta}{d t}(0)=0$$

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• Paper 3, Section II, A

  #Methods

  2011

  A uniform stretched string of length $L$, density per unit length $\mu$ and tension $T=\mu c^{2}$ is fixed at both ends. Its transverse displacement is given by $y(x, t)$ for $0 \leqslant x \leqslant L$. The motion of the string is resisted by the surrounding medium with a resistive force per unit length of $-2 k \mu \frac{\partial y}{\partial t}$.

  (i) Show that the equation of motion of the string is

  $$\frac{\partial^{2} y}{\partial t^{2}}+2 k \frac{\partial y}{\partial t}-c^{2} \frac{\partial^{2} y}{\partial x^{2}}=0$$

  provided that the transverse motion can be regarded as small.

  (ii) Suppose now that $k=\frac{\pi c}{L}$. Find the displacement of the string for $t \geqslant 0$ given the initial conditions

  $$y(x, 0)=A \sin \left(\frac{\pi x}{L}\right) \quad \text { and } \quad \frac{\partial y}{\partial t}(x, 0)=0$$

  (iii) Sketch the transverse displacement at $x=\frac{L}{2}$ as a function of time for $t \geqslant 0$.

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• Paper 4, Section II, A

  #Methods

  2011

  Let $D$ be a two dimensional domain with boundary $\partial D$. Establish Green's second identity

  $$\int_{D}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d A=\int_{\partial D}\left(\phi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \phi}{\partial n}\right) d s$$

  where $\frac{\partial}{\partial n}$ denotes the outward normal derivative on $\partial D$.

  State the differential equation and boundary conditions which are satisfied by a Dirichlet Green's function $G\left(\mathbf{r}, \mathbf{r}_{0}\right)$ for the Laplace operator on the domain $D$, where $\mathbf{r}_{0}$ is a fixed point in the interior of $D$.

  Suppose that $\nabla^{2} \psi=0$ on $D$. Show that

  $$\psi\left(\mathbf{r}_{0}\right)=\int_{\partial D} \psi(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{\mathbf{0}}\right) d s$$

  Consider Laplace's equation in the upper half plane,

  $$\nabla^{2} \psi(x, y)=0, \quad-\infty<x<\infty \quad \text { and } \quad y>0$$

  with boundary conditions $\psi(x, 0)=f(x)$ where $f(x) \rightarrow 0$ as $|x| \rightarrow \infty$, and $\psi(x, y) \rightarrow 0$ as $\sqrt{x^{2}+y^{2}} \rightarrow \infty$. Show that the solution is given by the integral formula

  $$\psi\left(x_{0}, y_{0}\right)=\frac{y_{0}}{\pi} \int_{-\infty}^{\infty} \frac{f(x)}{\left(x-x_{0}\right)^{2}+y_{0}^{2}} d x$$

  [ Hint: It might be useful to consider

  $$G\left(\mathbf{r}, \mathbf{r}_{0}\right)=\frac{1}{2 \pi}\left(\log \left|\mathbf{r}-\mathbf{r}_{0}\right|-\log \left|\mathbf{r}-\tilde{\mathbf{r}}_{0}\right|\right)$$

  for suitable $\tilde{\mathbf{r}}_{\mathbf{0}}$. You may assume $\nabla^{2} \log \left|\mathbf{r}-\mathbf{r}_{0}\right|=2 \pi \delta\left(\mathbf{r}-\mathbf{r}_{0}\right)$. ]

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• Paper 2, Section I, 4G

  #Metric and Topological Spaces

  2011

  (i) Let $t>0$. For $\mathbf{x}=(x, y), \mathbf{x}^{\prime}=\left(x^{\prime}, y^{\prime}\right) \in \mathbb{R}^{2}$, let

  $$\begin{gathered} d\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|x^{\prime}-x\right|+t\left|y^{\prime}-y\right|, \\ \delta\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\sqrt{\left(x^{\prime}-x\right)^{2}+\left(y^{\prime}-y\right)^{2}} \end{gathered}$$

  ( $\delta$ is the usual Euclidean metric on $\mathbb{R}^{2}$.) Show that $d$ is a metric on $\mathbb{R}^{2}$ and that the two metrics $d, \delta$ give rise to the same topology on $\mathbb{R}^{2}$.

  (ii) Give an example of a topology on $\mathbb{R}^{2}$, different from the one in (i), whose induced topology (subspace topology) on the $x$-axis is the usual topology (the one defined by the metric $\left.d\left(x, x^{\prime}\right)=\left|x^{\prime}-x\right|\right)$. Justify your answer.

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• Paper 3, Section I, 3G

  #Metric and Topological Spaces

  2011

  Let $X, Y$ be topological spaces, and suppose $Y$ is Hausdorff.

  (i) Let $f, g: X \rightarrow Y$ be two continuous maps. Show that the set

  $$E(f, g):=\{x \in X \mid f(x)=g(x)\} \subset X$$

  is a closed subset of $X$.

  (ii) Let $W$ be a dense subset of $X$. Show that a continuous map $f: X \rightarrow Y$ is determined by its restriction $\left.f\right|_{W}$ to $W$.

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• Paper 1, Section II, G

  #Metric and Topological Spaces

  2011

  Let $X$ be a metric space with the distance function $d: X \times X \rightarrow \mathbb{R}$. For a subset $Y$ of $X$, its diameter is defined as $\delta(Y):=\sup \left\{d\left(y, y^{\prime}\right) \mid y, y^{\prime} \in Y\right\}$.

  Show that, if $X$ is compact and $\left\{U_{\lambda}\right\}_{\lambda \in \Lambda}$ is an open covering of $X$, then there exists an $\epsilon>0$ such that every subset $Y \subset X$ with $\delta(Y)<\epsilon$ is contained in some $U_{\lambda}$.

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• Paper 4, Section II, 13G

  #Metric and Topological Spaces

  2011

  Let $X, Y$ be topological spaces and $X \times Y$ their product set. Let $p_{Y}: X \times Y \rightarrow Y$ be the projection map.

  (i) Define the product topology on $X \times Y$. Prove that if a subset $Z \subset X \times Y$ is open then $p_{Y}(Z)$ is open in $Y$.

  (ii) Give an example of $X, Y$ and a closed set $Z \subset X \times Y$ such that $p_{Y}(Z)$ is not closed.

  (iii) When $X$ is compact, show that if a subset $Z \subset X \times Y$ is closed then $p_{Y}(Z)$ is closed

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• Paper 1, Section I, B

  #Numerical Analysis

  2011

  Orthogonal monic polynomials $p_{0}, p_{1}, \ldots, p_{n}, \ldots$ are defined with respect to the inner product $\langle p, q\rangle=\int_{-1}^{1} w(x) p(x) q(x) d x$, where $p_{n}$ is of degree $n$. Show that such polynomials obey a three-term recurrence relation

  $$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x)$$

  for appropriate choices of $\alpha_{n}$ and $\beta_{n}$.

  Now suppose that $w(x)$ is an even function of $x$. Show that the $p_{n}$ are even or odd functions of $x$ according to whether $n$ is even or odd.

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• Paper 4, Section I, B

  #Numerical Analysis

  2011

  Consider the multistep method for numerical solution of the differential equation $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$ :

  $$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$

  What does it mean to say that the method is of order $p$, and that the method is convergent?

  Show that the method is of order $p$ if

  $$\sum_{l=0}^{s} \rho_{l}=0, \quad \sum_{l=0}^{s} l^{k} \rho_{l}=k \sum_{l=0}^{s} l^{k-1} \sigma_{l}, \quad k=1,2, \ldots, p$$

  and give the conditions on $\rho(w)=\sum_{l=0}^{s} \rho_{l} w^{l}$ that ensure convergence.

  Hence determine for what values of $\theta$ and the $\sigma_{i}$ the two-step method

  $$\mathbf{y}_{n+2}-(1-\theta) \mathbf{y}_{n+1}-\theta \mathbf{y}_{n}=h\left[\sigma_{0} \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\sigma_{1} \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)+\sigma_{2} \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)\right]$$

  is (a) convergent, and (b) of order 3 .

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• Paper 1, Section II, B

  #Numerical Analysis

  2011

  Consider a function $f(x)$ defined on the domain $x \in[0,1]$. Find constants $\alpha, \beta$, so that for any fixed $\xi \in[0,1]$,

  $$f^{\prime \prime}(\xi)=\alpha f(0)+\beta f^{\prime}(0)+\gamma f(1)$$

  is exactly satisfied for polynomials of degree less than or equal to two.

  By using the Peano kernel theorem, or otherwise, show that

  $$\begin{aligned} f^{\prime}(\xi)-f^{\prime}(0)-\xi(\alpha f(0)&\left.+\beta f^{\prime}(0)+\gamma f(1)\right)=\int_{0}^{\xi}(\xi-\theta) H_{1}(\theta) f^{\prime \prime \prime}(\theta) d \theta \\ &+\int_{0}^{\xi} \theta H_{2}(\theta) f^{\prime \prime \prime}(\theta) d \theta+\int_{\xi}^{1} \xi H_{2}(\theta) f^{\prime \prime \prime}(\theta) d \theta \end{aligned}$$

  where $H_{1}(\theta)=1-(1-\theta)^{2} \geqslant 0, H_{2}(\theta)=-(1-\theta)^{2} \leqslant 0$. Thus show that

  $$\left|f^{\prime}(\xi)-f^{\prime}(0)-\xi\left(\alpha f(0)+\beta f^{\prime}(0)+\gamma f(1)\right)\right| \leqslant\left.\frac{1}{6}\left(2 \xi-3 \xi^{2}+4 \xi^{3}-\xi^{4}\right)|| f^{\prime \prime \prime}\right|_{\infty} .$$

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• Paper 2, Section II, B

  #Numerical Analysis

  2011

  What is the $Q R$-decomposition of a matrix A? Explain how to construct the matrices $Q$ and $R$ by the Gram-Schmidt procedure, and show how the decomposition can be used to solve the matrix equation $A \mathbf{x}=\mathbf{b}$ when $A$ is a square matrix.

  Why is this procedure not useful for numerical decomposition of large matrices? Give a brief description of an alternative procedure using Givens rotations.

  Find a $Q R$-decomposition for the matrix

  $$\mathrm{A}=\left[\begin{array}{rrrr} 3 & 4 & 7 & 13 \\ -6 & -8 & -8 & -12 \\ 3 & 4 & 7 & 11 \\ 0 & 2 & 5 & 7 \end{array}\right]$$

  Is your decomposition unique? Use the decomposition you have found to solve the equation

  $$A x=\left[\begin{array}{c} 4 \\ 6 \\ 2 \\ 9 \end{array}\right]$$

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• Paper 3, Section II, B

  #Numerical Analysis

  2011

  A Gaussian quadrature formula provides an approximation to the integral

  $$\int_{-1}^{1}\left(1-x^{2}\right) f(x) d x \approx \sum_{k=1}^{\nu} b_{k} f\left(c_{k}\right)$$

  which is exact for all $f(x)$ that are polynomials of degree $\leqslant(2 \nu-1)$.

  Write down explicit expressions for the $b_{k}$ in terms of integrals, and explain why it is necessary that the $c_{k}$ are the zeroes of a (monic) polynomial $p_{\nu}$ of degree $\nu$ that satisfies $\int_{-1}^{1}\left(1-x^{2}\right) p_{\nu}(x) q(x) d x=0$ for any polynomial $q(x)$ of degree less than $\nu .$

  The first such polynomials are $p_{0}=1, p_{1}=x, p_{2}=x^{2}-1 / 5, p_{3}=x^{3}-3 x / 7$. Show that the Gaussian quadrature formulae for $\nu=2,3$ are

  $$\begin{array}{ll} \nu=2: & \frac{2}{3}\left[f\left(-\frac{1}{\sqrt{5}}\right)+f\left(\frac{1}{\sqrt{5}}\right)\right] \\ \nu=3: & \frac{14}{45}\left[f\left(-\sqrt{\frac{3}{7}}\right)+f\left(\sqrt{\frac{3}{7}}\right)\right]+\frac{32}{45} f(0) \end{array}$$

  Verify the result for $\nu=3$ by considering $f(x)=1, x^{2}, x^{4}$.

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• Paper 1, Section I, H

  #Optimization

  2011

  Suppose that $A x \leqslant b$ and $x \geqslant 0$ and $A^{T} y \geqslant c$ and $y \geqslant 0$ where $x$ and $c$ are $n$-dimensional column vectors, $y$ and $b$ are $m$-dimensional column vectors, and $A$ is an $m \times n$ matrix. Here, the vector inequalities are interpreted component-wise.

  (i) Show that $c^{T} x \leqslant b^{T} y$.

  (ii) Find the maximum value of

  $$\begin{aligned} 6 x_{1}+8 x_{2}+3 x_{3} \quad \text { subject to } & 2 x_{1}+4 x_{2}+x_{3} \leqslant 10 \\ & 3 x_{1}+4 x_{2}+3 x_{3} \leqslant 6 \\ & x_{1}, x_{2}, x_{3} \geqslant 0 \end{aligned}$$

  You should state any results from the course used in your solution.

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• Paper 2, Section I, H

  #Optimization

  2011

  Let $N=\{1, \ldots, n\}$ be the set of nodes of a network, where 1 is the source and $n$ is the $\operatorname{sink}$. Let $c_{i j}$ denote the capacity of the arc from node $i$ to node $j$.

  (i) In the context of maximising the flow through this network, define the following terms: feasible flow, flow value, cut, cut capacity.

  (ii) State and prove the max-flow min-cut theorem for network flows.

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• Paper 3, Section II, H

  #Optimization

  2011

  (i) What does it mean to say a set $C \subseteq \mathbb{R}^{n}$ is convex?

  (ii) What does it mean to say $z$ is an extreme point of a convex set $C ?$

  Let $A$ be an $m \times n$ matrix, where $n>m$. Let $b$ be an $m \times 1$ vector, and let

  $$C=\left\{x \in \mathbb{R}^{n}: A x=b, x \geqslant 0\right\}$$

  where the inequality is interpreted component-wise.

  (iii) Show that $C$ is convex.

  (iv) Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at least $m+1$ indices $i$ are such that $z_{i}>0$. Show that $z$ is not an extreme point of $C$. [Hint: If $r>m$, then any set of $r$ vectors in $\mathbb{R}^{m}$ is linearly dependent.]

  (v) Now suppose that every set of $m$ columns of $A$ is linearly independent. Let $z=\left(z_{1}, \ldots, z_{n}\right)^{T}$ be a point in $C$ with the property that at most $m$ indices $i$ are such that $z_{i}>0$. Show that $z$ is an extreme point of $C$.

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• Paper 4, Section II, H

  #Optimization

  2011

  A company must ship coal from four mines, labelled $A, B, C, D$, to supply three factories, labelled $a, b, c$. The per unit transport cost, the outputs of the mines, and the requirements of the factories are given below.

  \begin{tabular}{c|c|c|c|c|c} & $A$ & $B$ & $C$ & $D$ & \ \hline$a$ & 12 & 3 & 5 & 2 & 34 \ \hline$b$ & 4 & 11 & 2 & 6 & 21 \ \hline$c$ & 3 & 9 & 7 & 4 & 23 \ \hline & 20 & 32 & 15 & 11 & \end{tabular}

  For instance, mine $B$ can produce 32 units of coal, factory a requires 34 units of coal, and it costs 3 units of money to ship one unit of coal from $B$ to $a$. What is the minimal cost of transporting coal from the mines to the factories?

  Now suppose increased efficiency allows factory $b$ to reduce its requirement to $20.8$ units of coal, and as a consequence, mine $B$ reduces its output to $31.8$ units. By how much does the transport cost decrease?

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• Paper 3, Section I, C

  #Quantum Mechanics

  2011

  A particle of mass $m$ and energy $E$, incident from $x=-\infty$, scatters off a delta function potential at $x=0$. The time independent Schrödinger equation is

  $$-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi}{d x^{2}}+U \delta(x) \psi=E \psi$$

  where $U$ is a positive constant. Find the reflection and transmission probabilities.

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• Paper 4, Section $I$, C

  #Quantum Mechanics

  2011

  Consider the 3-dimensional oscillator with Hamiltonian

  $$H=-\frac{\hbar^{2}}{2 m} \nabla^{2}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}+4 z^{2}\right)$$

  Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels.

  [You may assume that the energy levels of the 1-dimensional harmonic oscillator with Hamiltonian

  $$H_{0}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{m \omega^{2}}{2} x^{2}$$

  $\left.\operatorname{are}\left(n+\frac{1}{2}\right) \hbar \omega, n=0,1,2, \ldots .\right]$

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• Paper 1, Section II, C

  #Quantum Mechanics

  2011

  For a quantum mechanical particle moving freely on a circle of length $2 \pi$, the wavefunction $\psi(t, x)$ satisfies the Schrödinger equation

  $$i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}}$$

  on the interval $0 \leqslant x \leqslant 2 \pi$, and also the periodicity conditions $\psi(t, 2 \pi)=\psi(t, 0)$, and $\frac{\partial \psi}{\partial x}(t, 2 \pi)=\frac{\partial \psi}{\partial x}(t, 0)$. Find the allowed energy levels of the particle, and their degeneracies.

  The current is defined as

  $$j=\frac{i \hbar}{2 m}\left(\psi{\frac{\partial \psi^{*}}{\partial x}}^{*}-\psi^{*} \frac{\partial \psi}{\partial x}\right)$$

  where $\psi$ is a normalized state. Write down the general normalized state of the particle when it has energy $2 \hbar^{2} / m$, and show that in any such state the current $j$ is independent of $x$ and $t$. Find a state with this energy for which the current has its maximum positive value, and find a state with this energy for which the current vanishes.

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• Paper 2, Section II, C

  #Quantum Mechanics

  2011

  The quantum mechanical angular momentum operators are

  $$L_{i}=-i \hbar \epsilon_{i j k} x_{j} \frac{\partial}{\partial x_{k}} \quad(i=1,2,3)$$

  Show that each of these is hermitian.

  The total angular momentum operator is defined as $\mathbf{L}^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}$. Show that $\left\langle\mathbf{L}^{2}\right\rangle \geqslant\left\langle L_{3}^{2}\right\rangle$ in any state, and show that the only states where $\left\langle\mathbf{L}^{2}\right\rangle=\left\langle L_{3}^{2}\right\rangle$ are those with no angular dependence. Verify that the eigenvalues of the operators $\mathbf{L}^{2}$ and $L_{3}^{2}$ (whose values you may quote without proof) are consistent with these results.

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• Paper 3, Section II, C

  #Quantum Mechanics

  2011

  For an electron in a hydrogen atom, the stationary state wavefunctions are of the form $\psi(r, \theta, \phi)=R(r) Y_{l m}(\theta, \phi)$, where in suitable units $R$ obeys the radial equation

  $$\frac{d^{2} R}{d r^{2}}+\frac{2}{r} \frac{d R}{d r}-\frac{l(l+1)}{r^{2}} R+2\left(E+\frac{1}{r}\right) R=0 .$$

  Explain briefly how the terms in this equation arise.

  This radial equation has bound state solutions of energy $E=E_{n}$, where $E_{n}=-\frac{1}{2 n^{2}}(n=1,2,3, \ldots)$. Show that when $l=n-1$, there is a solution of the form $R(r)=r^{\alpha} e^{-r / n}$, and determine $\alpha$. Find the expectation value $\langle r\rangle$ in this state.

  What is the total degeneracy of the energy level with energy $E_{n}$ ?

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• Paper 1, Section I, $\mathbf{7 H} \quad$

  #Statistics

  2011

  Consider the experiment of tossing a coin $n$ times. Assume that the tosses are independent and the coin is biased, with unknown probability $p$ of heads and $1-p$ of tails. A total of $X$ heads is observed.

  (i) What is the maximum likelihood estimator $\widehat{p}$ of $p$ ?

  Now suppose that a Bayesian statistician has the $\operatorname{Beta}(M, N)$ prior distribution for $p$.

  (ii) What is the posterior distribution for $p$ ?

  (iii) Assuming the loss function is $L(p, a)=(p-a)^{2}$, show that the statistician's point estimate for $p$ is given by

  $$\frac{M+X}{M+N+n}$$

  [The $\operatorname{Beta}(M, N)$ distribution has density $\frac{\Gamma(M+N)}{\Gamma(M) \Gamma(N)} x^{M-1}(1-x)^{N-1}$ for $0<x<1$ and $\left.\operatorname{mean} \frac{M}{M+N} .\right]$

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• Paper 2, Section I, H

  #Statistics

  2011

  Let $X_{1}, \ldots, X_{n}$ be random variables with joint density function $f\left(x_{1}, \ldots, x_{n} ; \theta\right)$, where $\theta$ is an unknown parameter. The null hypothesis $H_{0}: \theta=\theta_{0}$ is to be tested against the alternative hypothesis $H_{1}: \theta=\theta_{1}$.

  (i) Define the following terms: critical region, Type I error, Type II error, size, power.

  (ii) State and prove the Neyman-Pearson lemma.

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• Paper 1, Section II, H

  #Statistics

  2011

  Let $X_{1}, \ldots, X_{n}$ be independent random variables with probability mass function $f(x ; \theta)$, where $\theta$ is an unknown parameter.

  (i) What does it mean to say that $T$ is a sufficient statistic for $\theta$ ? State, but do not prove, the factorisation criterion for sufficiency.

  (ii) State and prove the Rao-Blackwell theorem.

  Now consider the case where $f(x ; \theta)=\frac{1}{x !}(-\log \theta)^{x} \theta$ for non-negative integer $x$ and $0<\theta<1$.

  (iii) Find a one-dimensional sufficient statistic $T$ for $\theta$.

  (iv) Show that $\tilde{\theta}=\mathbb{\prod}_{\left\{X_{1}=0\right\}}$ is an unbiased estimator of $\theta$.

  (v) Find another unbiased estimator $\widehat{\theta}$ which is a function of the sufficient statistic $T$ and that has smaller variance than $\tilde{\theta}$. You may use the following fact without proof: $X_{1}+\cdots+X_{n}$ has the Poisson distribution with parameter $-n \log \theta$.

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• Paper 3, Section II, H

  #Statistics

  2011

  Consider the general linear model

  $$Y=X \beta+\epsilon$$

  where $X$ is a known $n \times p$ matrix, $\beta$ is an unknown $p \times 1$ vector of parameters, and $\epsilon$ is an $n \times 1$ vector of independent $N\left(0, \sigma^{2}\right)$ random variables with unknown variance $\sigma^{2}$. Assume the $p \times p$ matrix $X^{T} X$ is invertible.

  (i) Derive the least squares estimator $\widehat{\beta}$ of $\beta$.

  (ii) Derive the distribution of $\widehat{\beta}$. Is $\widehat{\beta}$ an unbiased estimator of $\beta$ ?

  (iii) Show that $\frac{1}{\sigma^{2}}\|Y-X \widehat{\beta}\|^{2}$ has the $\chi^{2}$ distribution with $k$ degrees of freedom, where $k$ is to be determined.

  (iv) Let $\tilde{\beta}$ be an unbiased estimator of $\beta$ of the form $\tilde{\beta}=C Y$ for some $p \times n$ matrix $C$. By considering the matrix $\mathbb{E}\left[(\widehat{\beta}-\widetilde{\beta})(\widehat{\beta}-\beta)^{T}\right]$ or otherwise, show that $\widehat{\beta}$ and $\widehat{\beta}-\widetilde{\beta}$ are independent.

  [You may use standard facts about the multivariate normal distribution as well as results from linear algebra, including the fact that $I-X\left(X^{T} X\right)^{-1} X^{T}$ is a projection matrix of rank $n-p$, as long as they are carefully stated.]

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• Paper 4, Section II, H

  #Statistics

  2011

  Consider independent random variables $X_{1}, \ldots, X_{n}$ with the $N\left(\mu_{X}, \sigma_{X}^{2}\right)$ distribution and $Y_{1}, \ldots, Y_{n}$ with the $N\left(\mu_{Y}, \sigma_{Y}^{2}\right)$ distribution, where the means $\mu_{X}, \mu_{Y}$ and variances $\sigma_{X}^{2}, \sigma_{Y}^{2}$ are unknown. Derive the generalised likelihood ratio test of size $\alpha$ of the null hypothesis $H_{0}: \sigma_{X}^{2}=\sigma_{Y}^{2}$ against the alternative $H_{1}: \sigma_{X}^{2} \neq \sigma_{Y}^{2}$. Express the critical region in terms of the statistic $T=\frac{S_{X X}}{S_{X X}+S_{Y Y}}$ and the quantiles of a beta distribution, where

  $$S_{X X}=\sum_{i=1}^{n} X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} X_{i}\right)^{2} \text { and } S_{Y Y}=\sum_{i=1}^{n} Y_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} Y_{i}\right)^{2}$$

  [You may use the following fact: if $U \sim \Gamma(a, \lambda)$ and $V \sim \Gamma(b, \lambda)$ are independent, then $\left.\frac{U}{U+V} \sim \operatorname{Beta}(a, b) .\right]$

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• Paper 1, Section I, D

  #Variational Principles

  2011

  (i) Write down the Euler-Lagrange equations for the volume integral

  $$\int_{V}(\nabla u \cdot \nabla u+12 u) d V$$

  where $V$ is the unit ball $x^{2}+y^{2}+z^{2} \leqslant 1$, and verify that the function $u(x, y, z)=x^{2}+y^{2}+z^{2}$ gives a stationary value of the integral subject to the condition $u=1$ on the boundary.

  (ii) Write down the Euler-Lagrange equations for the integral

  $$\int_{0}^{1}\left(\dot{x}^{2}+\dot{y}^{2}+4 x+4 y\right) d t$$

  where the dot denotes differentiation with respect to $t$, and verify that the functions $x(t)=t^{2}, y(t)=t^{2}$ give a stationary value of the integral subject to the boundary conditions $x(0)=y(0)=0$ and $x(1)=y(1)=1$.

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• Paper 3, Section I, D

  #Variational Principles

  2011

  Find, using a Lagrange multiplier, the four stationary points in $\mathbb{R}^{3}$ of the function $x^{2}+y^{2}+z^{2}$ subject to the constraint $x^{2}+2 y^{2}-z^{2}=1$. By considering the situation geometrically, or otherwise, identify the nature of the constrained stationary points.

  How would your answers differ if, instead, the stationary points of the function $x^{2}+2 y^{2}-z^{2}$ were calculated subject to the constraint $x^{2}+y^{2}+z^{2}=1 ?$

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• Paper 2, Section II, D

  #Variational Principles

  2011

  (i) Let $I[y]=\int_{0}^{1}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y$ is twice differentiable and $y(0)=y(1)=0$. Write down the associated Euler-Lagrange equation and show that the only solution is $y(x)=0$.

  (ii) Let $J[y]=\int_{0}^{1}\left(y^{\prime}+y \tan x\right)^{2} d x$, where $y$ is twice differentiable and $y(0)=y(1)=$ 0 . Show that $J[y]=0$ only if $y(x)=0$.

  (iii) Show that $I[y]=J[y]$ and deduce that the extremal value of $I[y]$ is a global minimum.

  (iv) Use the second variation of $I[y]$ to verify that the extremal value of $I[y]$ is a local minimum.

  (v) How would your answers to part (i) differ in the case $I[y]=\int_{0}^{2 \pi}\left(\left(y^{\prime}\right)^{2}-y^{2}\right) d x$, where $y(0)=y(2 \pi)=0$ ? Show that the solution $y(x)=0$ is not a global minimizer in this case. (You may use without proof the result $I[x(2 \pi-x)]=-\frac{8}{15}\left(2 \pi^{2}-5\right)$.) Explain why the arguments of parts (iii) and (iv) cannot be used.

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• Paper 1, Section II, H

  #Algebraic Geometry

  2011

  (i) Let $X$ be an affine variety over an algebraically closed field. Define what it means for $X$ to be irreducible, and show that if $U$ is a non-empty open subset of an irreducible $X$, then $U$ is dense in $X$.

  (ii) Show that $n \times n$ matrices with distinct eigenvalues form an affine variety, and are a Zariski open subvariety of affine space $\mathbb{A}^{n^{2}}$ over an algebraically closed field.

  (iii) Let $\operatorname{char}_{A}(x)=\operatorname{det}(x I-A)$ be the characteristic polynomial of $A$. Show that the $n \times n$ matrices $A$ such that $\operatorname{char}_{A}(A)=0$ form a Zariski closed subvariety of $\mathbb{A}^{n^{2}}$. Hence conclude that this subvariety is all of $\mathbb{A}^{n^{2}}$.

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• Paper 2, Section II, H

  #Algebraic Geometry

  2011

  (i) Let $k$ be an algebraically closed field, and let $I$ be an ideal in $k\left[x_{0}, \ldots, x_{n}\right]$. Define what it means for $I$ to be homogeneous.

  Now let $Z \subseteq \mathbb{A}^{n+1}$ be a Zariski closed subvariety invariant under $k^{*}=k-\{0\}$; that is, if $z \in Z$ and $\lambda \in k^{*}$, then $\lambda z \in Z$. Show that $I(Z)$ is a homogeneous ideal.

  (ii) Let $f \in k\left[x_{1}, \ldots, x_{n-1}\right]$, and let $\Gamma=\left\{(x, f(x)) \mid x \in \mathbb{A}^{n-1}\right\} \subseteq \mathbb{A}^{n}$ be the graph of $f$.

  Let $\bar{\Gamma}$ be the closure of $\Gamma$ in $\mathbb{P}^{n}$.

  Write, in terms of $f$, the homogeneous equations defining $\bar{\Gamma}$.

  Assume that $k$ is an algebraically closed field of characteristic zero. Now suppose $n=3$ and $f(x, y)=y^{3}-x^{2} \in k[x, y]$. Find the singular points of the projective surface $\bar{\Gamma}$.

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• Paper 3, Section II, H

  #Algebraic Geometry

  2011

  Let $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic 0 .

  (i) Let $D$ be a divisor on $X$.

  Define $\mathcal{L}(D)$, and show $\operatorname{dim} \mathcal{L}(D) \leqslant \operatorname{deg} D+1$.

  (ii) Define the space of rational differentials $\Omega_{k(X) / k}^{1}$.

  If $p$ is a point on $X$, and $t$ a local parameter at $p$, show that $\Omega_{k(X) / k}^{1}=k(X) d t$.

  Use that equality to give a definition of $v_{p}(\omega) \in \mathbb{Z}$, for $\omega \in \Omega_{k(X) / k}^{1}, p \in X$. [You need not show that your definition is independent of the choice of local parameter.]

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• Paper 4, Section II, H

  #Algebraic Geometry

  2011

  Let $X$ be a smooth projective curve over an algebraically closed field $k$.

  State the Riemann-Roch theorem, briefly defining all the terms that appear.

  Now suppose $X$ has genus 1 , and let $P_{\infty} \in X$.

  Compute $\mathcal{L}\left(n P_{\infty}\right)$ for $n \leqslant 6$. Show that $\phi_{3 P_{\infty}}$ defines an isomorphism of $X$ with a smooth plane curve in $\mathbb{P}^{2}$ which is defined by a polynomial of degree 3 .

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• Paper 1, Section II, H

  #Algebraic Topology

  2011

  Are the following statements true or false? Justify your answers.

  (i) If $x$ and $y$ lie in the same path-component of $X$, then $\Pi_{1}(X, x) \cong \Pi_{1}(X, y)$.

  (ii) If $x$ and $y$ are two points of the Klein bottle $K$, and $u$ and $v$ are two paths from $x$ to $y$, then $u$ and $v$ induce the same isomorphism from $\Pi_{1}(K, x)$ to $\Pi_{1}(K, y)$.

  (iii) $\Pi_{1}(X \times Y,(x, y))$ is isomorphic to $\Pi_{1}(X, x) \times \Pi_{1}(Y, y)$ for any two spaces $X$ and $Y$.

  (iv) If $X$ and $Y$ are connected polyhedra and $H_{1}(X) \cong H_{1}(Y)$, then $\Pi_{1}(X) \cong \Pi_{1}(Y)$.

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• Paper 2, Section II, H

  #Algebraic Topology

  2011

  Explain what is meant by a covering projection. State and prove the pathlifting property for covering projections, and indicate briefly how it generalizes to a lifting property for homotopies between paths. [You may assume the Lebesgue Covering Theorem.]

  Let $X$ be a simply connected space, and let $G$ be a subgroup of the group of all homeomorphisms $X \rightarrow X$. Suppose that, for each $x \in X$, there exists an open neighbourhood $U$ of $x$ such that $U \cap g[U]=\emptyset$ for each $g \in G$ other than the identity. Show that the projection $p: X \rightarrow X / G$ is a covering projection, and deduce that $\Pi_{1}(X / G) \cong G$.

  By regarding $S^{3}$ as the set of all quaternions of modulus 1 , or otherwise, show that there is a quotient space of $S^{3}$ whose fundamental group is a non-abelian group of order $8 .$

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• Paper 3, Section II, H

  #Algebraic Topology

  2011

  Let $K$ and $L$ be (finite) simplicial complexes. Explain carefully what is meant by a simplicial approximation to a continuous map $f:|K| \rightarrow|L|$. Indicate briefly how the cartesian product $|K| \times|L|$ may be triangulated.

  Two simplicial maps $g, h: K \rightarrow L$ are said to be contiguous if, for each simplex $\sigma$ of $K$, there exists a simplex $\sigma *$ of $L$ such that both $g(\sigma)$ and $h(\sigma)$ are faces of $\sigma *$. Show that:

  (i) any two simplicial approximations to a given map $f:|K| \rightarrow|L|$ are contiguous;

  (ii) if $g$ and $h$ are contiguous, then they induce homotopic maps $|K| \rightarrow|L|$;

  (iii) if $f$ and $g$ are homotopic maps $|K| \rightarrow|L|$, then for some subdivision $K^{(n)}$ of $K$ there exists a sequence $\left(h_{1}, h_{2}, \ldots, h_{m}\right)$ of simplicial maps $K^{(n)} \rightarrow L$ such that $h_{1}$ is a simplicial approximation to $f, h_{m}$ is a simplicial approximation to $g$ and each pair $\left(h_{i}, h_{i+1}\right)$ is contiguous.

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• Paper 4, Section II, H

  #Algebraic Topology

  2011

  State the Mayer-Vietoris theorem, and use it to calculate, for each integer $q>1$, the homology group of the space $X_{q}$ obtained from the unit disc $B^{2} \subseteq \mathbb{C}$ by identifying pairs of points $\left(z_{1}, z_{2}\right)$ on its boundary whenever $z_{1}^{q}=z_{2}^{q}$. [You should construct an explicit triangulation of $X_{q}$.]

  Show also how the theorem may be used to calculate the homology groups of the suspension $S K$ of a connected simplicial complex $K$ in terms of the homology groups of $K$, and of the wedge union $X \vee Y$ of two connected polyhedra. Hence show that, for any finite sequence $\left(G_{1}, G_{2}, \ldots, G_{n}\right)$ of finitely-generated abelian groups, there exists a polyhedron $X$ such that $H_{0}(X) \cong \mathbb{Z}, H_{i}(X) \cong G_{i}$ for $1 \leqslant i \leqslant n$ and $H_{i}(X)=0$ for $i>n$. [You may assume the structure theorem which asserts that any finitely-generated abelian group is isomorphic to a finite direct sum of (finite or infinite) cyclic groups.]

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• Paper 3, Section II, E

  #Applications of Quantum Mechanics

  2011

  An electron of mass $m$ moves in a $D$-dimensional periodic potential that satisfies the periodicity condition

  $$V(\boldsymbol{r}+\boldsymbol{l})=V(\boldsymbol{r}) \quad \forall l \in \Lambda,$$

  where $\Lambda$ is a D-dimensional Bravais lattice. State Bloch's theorem for the energy eigenfunctions of the electron.

  For a one-dimensional potential $V(x)$ such that $V(x+a)=V(x)$, give a full account of how the "nearly free electron model" leads to a band structure for the energy levels.

  Explain briefly the idea of a Fermi surface and its rôle in explaining the existence of conductors and insulators.

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• Paper 4, Section II, E

  #Applications of Quantum Mechanics

  2011

  A particle of charge $-e$ and mass $m$ moves in a magnetic field $\boldsymbol{B}(\boldsymbol{x}, t)$ and in an electric potential $\phi(\boldsymbol{x}, t)$. The time-dependent Schrödinger equation for the particle's wavefunction $\Psi(\boldsymbol{x}, t)$ is

  $$i \hbar\left(\frac{\partial}{\partial t}-\frac{i e}{\hbar} \phi\right) \Psi=-\frac{\hbar^{2}}{2 m}\left(\nabla+\frac{i e}{\hbar} \boldsymbol{A}\right)^{2} \Psi$$

  where $\boldsymbol{A}$ is the vector potential with $\boldsymbol{B}=\boldsymbol{\nabla} \wedge \boldsymbol{A}$. Show that this equation is invariant under the gauge transformations

  $$\begin{array}{ll} \boldsymbol{A}(\boldsymbol{x}, t) & \rightarrow \boldsymbol{A}(\boldsymbol{x}, t)+\boldsymbol{\nabla} f(\boldsymbol{x}, t) \\ \phi(\boldsymbol{x}, t) & \rightarrow \quad \phi(\boldsymbol{x}, t)-\frac{\partial}{\partial t} f(\boldsymbol{x}, t) \end{array}$$

  where $f$ is an arbitrary function, together with a suitable transformation for $\Psi$ which should be stated.

  Assume now that $\partial \Psi / \partial z=0$, so that the particle motion is only in the $x$ and $y$ directions. Let $\boldsymbol{B}$ be the constant field $\boldsymbol{B}=(0,0, B)$ and let $\phi=0$. In the gauge where $\boldsymbol{A}=(-B y, 0,0)$ show that the stationary states are given by

  $$\Psi_{k}(\boldsymbol{x}, t)=\psi_{k}(\boldsymbol{x}) e^{-i E t / \hbar}$$

  with

  $$\psi_{k}(\boldsymbol{x})=e^{i k x} \chi_{k}(y)$$

  Show that $\chi_{k}(y)$ is the wavefunction for a simple one-dimensional harmonic oscillator centred at position $y_{0}=\hbar k / e B$. Deduce that the stationary states lie in infinitely degenerate levels (Landau levels) labelled by the integer $n \geqslant 0$, with energy

  $$E_{n}=(2 n+1) \frac{\hbar e B}{2 m}$$

  A uniform electric field $\mathcal{E}$ is applied in the $y$-direction so that $\phi=-\mathcal{E} y$. Show that the stationary states are given by $(*)$, where $\chi_{k}(y)$ is a harmonic oscillator wavefunction centred now at

  $$y_{0}=\frac{1}{e B}\left(\hbar k-m \frac{\mathcal{E}}{B}\right)$$

  Show also that the eigen-energies are given by

  $$E_{n, k}=(2 n+1) \frac{\hbar e B}{2 m}+e \mathcal{E} y_{0}+\frac{m \mathcal{E}^{2}}{2 B^{2}} .$$

  Why does this mean that the Landau energy levels are no longer degenerate in two dimensions?

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• Paper 1, Section II, E

  #Applications of Quantum Mechanics

  2011

  In one dimension a particle of mass $m$ and momentum $\hbar k, k>0$, is scattered by a potential $V(x)$ where $V(x) \rightarrow 0$ as $|x| \rightarrow \infty$. Incoming and outgoing plane waves of positive $(+)$ and negative $(-)$ parity are given, respectively, by

  $$\begin{array}{ll} I_{+}(k, x)=e^{-i k|x|}, & I_{-}(k, x)=\operatorname{sgn}(x) e^{-i k|x|} \\ O_{+}(k, x)=e^{i k|x|}, & O_{-}(k, x)=-\operatorname{sgn}(x) e^{i k|x|} \end{array}$$

  The scattering solutions to the time-independent Schrödinger equation with positive and negative parity incoming waves are $\psi_{+}(x)$ and $\psi_{-}(x)$, respectively. State how the asymptotic behaviour of $\psi_{+}$and $\psi_{-}$can be expressed in terms of $I_{+}, I_{-}, O_{+}, O_{-}$and the S-matrix denoted by

  $$\boldsymbol{S}=\left(\begin{array}{cc} S_{++} & S_{+-} \\ S_{-+} & S_{--} \end{array}\right)$$

  In the case where $V(x)=V(-x)$ explain briefly why you expect $S_{+-}=S_{-+}=0$.

  The potential $V(x)$ is given by

  $$V(x)=V_{0}[\delta(x-a)+\delta(x+a)]$$

  where $V_{0}$ is a constant. In this case, show that

  $$S_{--}(k)=e^{-2 i k a}\left[\frac{\left(2 k-i U_{0}\right) e^{i k a}+i U_{0} e^{-i k a}}{\left(2 k+i U_{0}\right) e^{-i k a}-i U_{0} e^{i k a}}\right]$$

  where $U_{0}=2 m V_{0} / \hbar^{2}$. Verify that $\left|S_{--}\right|^{2}=1$ and explain briefly the physical meaning of this result.

  For $V_{0}<0$, by considering the poles or zeros of $S_{--}(k)$ show that there exists one bound state of negative parity in this potential if $U_{0} a<-1$.

  For $V_{0}>0$ and $U_{0} a \gg 1$, show that $S_{--}(k)$ has a pole at

  $$k a=\pi+\alpha-i \gamma$$

  where, to leading order in $1 /\left(U_{0} a\right)$,

  $$\alpha=-\frac{\pi}{U_{0} a}, \quad \gamma=\left(\frac{\pi}{U_{0} a}\right)^{2}$$

  Explain briefy the physical meaning of this result, and why you expect that $\gamma>0$.

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• Paper 2, Section II, E

  #Applications of Quantum Mechanics

  2011

  A beam of particles of mass $m$ and momentum $p=\hbar k$, incident along the $z$-axis, is scattered by a spherically symmetric potential $V(r)$, where $V(r)=0$ for large $r$. State the boundary conditions on the wavefunction as $r \rightarrow \infty$ and hence define the scattering amplitude $f(\theta)$, where $\theta$ is the scattering angle.

  Given that, for large $r$,

  $$e^{i k r \cos \theta}=\frac{1}{2 i k r} \sum_{l=0}^{\infty}(2 l+1)\left(e^{i k r}-(-1)^{l} e^{-i k r}\right) P_{l}(\cos \theta)$$

  explain how the partial-wave expansion can be used to define the phase shifts $\delta_{l}(k)(l=$ $0,1,2, \ldots)$. Furthermore, given that $d \sigma / d \Omega=|f(\theta)|^{2}$, derive expressions for $f(\theta)$ and the total cross-section $\sigma$ in terms of the $\delta_{l}$.

  In a particular case $V(r)$ is given by

  $$V(r)=\left\{\begin{array}{cl} \infty, & r<a \\ -V_{0}, & a<r<2 a \\ 0, & r>2 a \end{array}\right.$$

  where $V_{0}>0$. Show that the $\mathrm{S}$-wave phase shift $\delta_{0}$ satisfies

  $$\tan \left(\delta_{0}\right)=\frac{k \cos (2 k a)-\kappa \cot (\kappa a) \sin (2 k a)}{k \sin (2 k a)+\kappa \cot (\kappa a) \cos (2 k a)},$$

  where $\kappa^{2}=2 m V_{0} / \hbar^{2}+k^{2}$.

  Derive an expression for the scattering length $a_{s}$ in terms of $\kappa$. Find the values of $\kappa$ for which $\left|a_{s}\right|$ diverges and briefly explain their physical significance.

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• Paper 1, Section II, J

  #Applied Probability

  2011

  (i) Let $X$ be a Markov chain with finitely many states. Define a stopping time and state the strong Markov property.

  (ii) Let $X$ be a Markov chain with state-space $\{-1,0,1\}$ and Q-matrix

  $$Q=\left(\begin{array}{ccc} -(q+\lambda) & \lambda & q \\ 0 & 0 & 0 \\ q & \lambda & -(q+\lambda) \end{array}\right), \text { where } q, \lambda>0$$

  Consider the integral $\int_{0}^{t} X(s) \mathrm{d} s$, the signed difference between the times spent by the chain at states $+1$ and $-1$ by time $t$, and let

  $$\begin{aligned} Y &=\sup \left[\int_{0}^{t} X(s) \mathrm{d} s: t>0\right] \\ \psi_{\pm}(c) &=\mathbb{P}\left(Y>c \mid X_{0}=\pm 1\right), \quad c>0 \end{aligned}$$

  Derive the equation

  $$\psi_{-}(c)=\int_{0}^{\infty} q e^{-(\lambda+q) u_{+}} \psi_{+}(c+u) \mathrm{d} u$$

  (iii) Obtain another equation relating $\psi_{+}$to $\psi_{-}$.

  (iv) Assuming that $\psi_{+}(c)=e^{-c A}, c>0$, where $A$ is a non-negative constant, calculate $A$.

  (v) Give an intuitive explanation why the function $\psi_{+}$must have the exponential form $\psi_{+}(c)=e^{-c A}$ for some $A$.

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• Paper 2, Section II, J

  #Applied Probability

  2011

  (i) Explain briefly what is meant by saying that a continuous-time Markov chain $X(t)$ is a birth-and-death process with birth rates $\lambda_{i}>0, i \geqslant 0$, and death rates $\mu_{i}>0$, $i \geqslant 1$.

  (ii) In the case where $X(t)$ is recurrent, find a sufficient condition on the birth and death parameters to ensure that

  $$\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j)=\pi_{j}>0, \quad j \geqslant 0$$

  and express $\pi_{j}$ in terms of these parameters. State the reversibility property of $X(t)$.

  Jobs arrive according to a Poisson process of rate $\lambda>0$. They are processed individually, by a single server, the processing times being independent random variables, each with the exponential distribution of rate $\nu>0$. After processing, the job either leaves the system, with probability $p, 0<p<1$, or, with probability $1-p$, it splits into two separate jobs which are both sent to join the queue for processing again. Let $X(t)$ denote the number of jobs in the system at time $t$.

  (iii) In the case $1+\lambda / \nu<2 p$, evaluate $\lim _{t \rightarrow \infty} \mathbb{P}(X(t)=j), j=0,1, \ldots$, and find the expected time that the processor is busy between two successive idle periods.

  (iv) What happens if $1+\lambda / \nu \geqslant 2 p$ ?

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• Paper 3, Section II, J

  #Applied Probability

  2011

  (i) Define an inhomogeneous Poisson process with rate function $\lambda(u)$.

  (ii) Show that the number of arrivals in an inhomogeneous Poisson process during the interval $(0, t)$ has the Poisson distribution with mean

  $$\int_{0}^{t} \lambda(u) \mathrm{d} u$$

  (iii) Suppose that $\Lambda=\{\Lambda(t), t \geqslant 0\}$ is a non-negative real-valued random process. Conditional on $\Lambda$, let $N=\{N(t), t \geqslant 0\}$ be an inhomogeneous Poisson process with rate function $\Lambda(u)$. Such a process $N$ is called a doubly-stochastic Poisson process. Show that the variance of $N(t)$ cannot be less than its mean.

  (iv) Now consider the process $M(t)$ obtained by deleting every odd-numbered point in an ordinary Poisson process of rate $\lambda$. Check that

  $$\mathbb{E} M(t)=\frac{2 \lambda t+e^{-2 \lambda t}-1}{4}, \quad \operatorname{Var} M(t)=\frac{4 \lambda t-8 \lambda t e^{-2 \lambda t}-e^{-4 \lambda t}+1}{16}$$

  Deduce that $M(t)$ is not a doubly-stochastic Poisson process.

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• Paper 4, Section II, J

  #Applied Probability

  2011

  At an $\mathrm{M} / \mathrm{G} / 1$ queue, the arrival times form a Poisson process of rate $\lambda$ while service times $S_{1}, S_{2}, \ldots$ are independent of each other and of the arrival times and have a common distribution $G$ with mean $\mathbb{E} S_{1}<+\infty$.

  (i) Show that the random variables $Q_{n}$ giving the number of customers left in the queue at departure times form a Markov chain.

  (ii) Specify the transition probabilities of this chain as integrals in $\mathrm{d} G(t)$ involving parameter $\lambda$. [No proofs are needed.]

  (iii) Assuming that $\rho=\lambda \mathbb{E} S_{1}<1$ and the chain $\left(Q_{n}\right)$ is positive recurrent, show that its stationary distribution $\left(\pi_{k}, k \geqslant 0\right)$ has the generating function given by

  $$\sum_{k \geqslant 0} \pi_{k} s^{k}=\frac{(1-\rho)(s-1) g(s)}{s-g(s)},|s| \leqslant 1$$

  for an appropriate function $g$, to be specified.

  (iv) Deduce that, in equilibrium, $Q_{n}$ has the mean value

  $$\rho+\frac{\lambda^{2} \mathbb{E} S_{1}^{2}}{2(1-\rho)}$$

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• Paper 1, Section II, A

  #Asymptotic Methods

  2011

  A function $f(n)$, defined for positive integer $n$, has an asymptotic expansion for large $n$ of the following form:

  $$f(n) \sim \sum_{k=0}^{\infty} a_{k} \frac{1}{n^{2 k}}, \quad n \rightarrow \infty$$

  What precisely does this mean?

  Show that the integral

  $$I(n)=\int_{0}^{2 \pi} \frac{\cos n t}{1+t^{2}} d t$$

  has an asymptotic expansion of the form $(*)$. [The Riemann-Lebesgue lemma may be used without proof.] Evaluate the coefficients $a_{0}, a_{1}$ and $a_{2}$.

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• Paper 3, Section II, A

  #Asymptotic Methods

  2011

  Let

  $$I_{0}=\int_{C_{0}} e^{x \phi(z)} d z \text {, }$$

  where $\phi(z)$ is a complex analytic function and $C_{0}$ is a steepest descent contour from a simple saddle point of $\phi(z)$ at $z_{0}$. Establish the following leading asymptotic approximation, for large real $x$ :

  $$I_{0} \sim i \sqrt{\frac{\pi}{2 \phi^{\prime \prime}\left(z_{0}\right) x}} e^{x \phi\left(z_{0}\right)}$$

  Let $n$ be a positive integer, and let

  $$I=\int_{C} e^{-t^{2}-2 n \ln t} d t$$

  where $C$ is a contour in the upper half $t$-plane connecting $t=-\infty$ to $t=\infty$, and $\ln t$ is real on the positive $t$-axis with a branch cut along the negative $t$-axis. Using the method of steepest descent, find the leading asymptotic approximation to $I$ for large $n$.

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• Paper 4, Section II, A

  #Asymptotic Methods

  2011

  Determine the range of the integer $n$ for which the equation

  $$\frac{d^{2} y}{d z^{2}}=z^{n} y$$

  has an essential singularity at $z=\infty$.

  Use the Liouville-Green method to find the leading asymptotic approximation to two independent solutions of

  $$\frac{d^{2} y}{d z^{2}}=z^{3} y$$

  for large $|z|$. Find the Stokes lines for these approximate solutions. For what range of $\arg z$ is the approximate solution which decays exponentially along the positive $z$-axis an asymptotic approximation to an exact solution with this exponential decay?

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• Paper 1, Section I, C

  #Classical Dynamics

  2011

  (i) A particle of mass $m$ and charge $q$, at position $\mathbf{x}$, moves in an electromagnetic field with scalar potential $\phi(\mathbf{x}, t)$ and vector potential $\mathbf{A}(\mathbf{x}, t)$. Verify that the Lagrangian

  $$L=\frac{1}{2} m \dot{\mathbf{x}}^{2}-q(\phi-\dot{\mathbf{x}} \cdot \mathbf{A})$$

  gives the correct equations of motion.

  [Note that $\mathbf{E}=-\nabla \phi-\dot{\mathbf{A}}$ and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$.]

  (ii) Consider the case of a constant uniform magnetic field, with $\mathbf{E}=\mathbf{0}$, given by $\phi=0$, $\mathbf{A}=(0, x B, 0)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant. Find the motion of the particle, and describe it carefully.

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• Paper 2, Section I, C

  #Classical Dynamics

  2011

  Three particles, each of mass $m$, move along a straight line. Their positions on the line containing the origin, $O$, are $x_{1}, x_{2}$ and $x_{3}$. They are subject to forces derived from the potential energy function

  $$V=\frac{1}{2} m \Omega^{2}\left[\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\left(x_{3}-x_{1}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right]$$

  Obtain Lagrange's equations for the system, and show that the frequency, $\omega$, of a normal mode satisfies

  $$f^{3}-9 f^{2}+24 f-16=0$$

  where $f=\left(\omega^{2} / \Omega^{2}\right)$. Find a complete set of normal modes for the system, and draw a diagram indicating the nature of the corresponding motions.

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• Paper 3, Section I, $\mathbf{9 C}$

  #Classical Dynamics

  2011

  The Lagrangian for a heavy symmetric top is

  $$L=\frac{1}{2} I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2} \sin ^{2} \theta\right)+\frac{1}{2} I_{3}(\dot{\psi}+\dot{\phi} \cos \theta)^{2}-M g l \cos \theta$$

  State Noether's Theorem. Hence, or otherwise, find two conserved quantities linear in momenta, and a third conserved quantity quadratic in momenta.

  Writing $\mu=\cos \theta$, deduce that $\mu$ obeys an equation of the form

  $$\dot{\mu}^{2}=F(\mu)$$

  where $F(\mu)$ is cubic in $\mu$. [You need not determine the explicit form of $F(\mu) .$ ]

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• Paper 4, Section I, C

  #Classical Dynamics

  2011

  (i) A dynamical system is described by the Hamiltonian $H\left(q_{i}, p_{i}\right)$. Define the Poisson bracket $\{f, g\}$ of two functions $f\left(q_{i}, p_{i}, t\right), g\left(q_{i}, p_{i}, t\right)$. Assuming the Hamiltonian equations of motion, find an expression for $d f / d t$ in terms of the Poisson bracket.

  (ii) A one-dimensional system has the Hamiltonian

  $$H=p^{2}+\frac{1}{q^{2}}$$

  Show that $u=p q-2 H t$ is a constant of the motion. Deduce the form of $(q(t), p(t))$ along a classical path, in terms of the constants $u$ and $H$.

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• Paper 2, Section II, C

  #Classical Dynamics

  2011

  Derive Euler's equations governing the torque-free and force-free motion of a rigid body with principal moments of inertia $I_{1}, I_{2}$ and $I_{3}$, where $I_{1}<I_{2}<I_{3}$. Identify two constants of the motion. Hence, or otherwise, find the equilibrium configurations such that the angular-momentum vector, as measured with respect to axes fixed in the body, remains constant. Discuss the stability of these configurations.

  A spacecraft may be regarded as moving in a torque-free and force-free environment. Nevertheless, flexing of various parts of the frame can cause significant dissipation of energy. How does the angular-momentum vector ultimately align itself within the body?

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• Paper 4, Section II, C

  #Classical Dynamics

  2011

  Given a Hamiltonian system with variables $\left(q_{i}, p_{i}\right), i=1, \ldots, n$, state the definition of a canonical transformation

  $$\left(q_{i}, p_{i}\right) \rightarrow\left(Q_{i}, P_{i}\right),$$

  where $\mathbf{Q}=\mathbf{Q}(\mathbf{q}, \mathbf{p}, t)$ and $\mathbf{P}=\mathbf{P}(\mathbf{q}, \mathbf{p}, t)$. Write down a matrix equation that is equivalent to the condition that the transformation is canonical.

  Consider a harmonic oscillator of unit mass, with Hamiltonian

  $$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right) .$$

  Write down the Hamilton-Jacobi equation for Hamilton's principal function $S(q, E, t)$, and deduce the Hamilton-Jacobi equation

  $$\frac{1}{2}\left[\left(\frac{\partial W}{\partial q}\right)^{2}+\omega^{2} q^{2}\right]=E$$

  for Hamilton's characteristic function $W(q, E)$.

  Solve (1) to obtain an integral expression for $W$, and deduce that, at energy $E$,

  $$S=\sqrt{2 E} \int d q \sqrt{\left(1-\frac{\omega^{2} q^{2}}{2 E}\right)}-E t$$

  Let $\alpha=E$, and define the angular coordinate

  $$\beta=\left(\frac{\partial S}{\partial E}\right)_{q, t}$$

  You may assume that (2) implies

  $$t+\beta=\left(\frac{1}{\omega}\right) \arcsin \left(\frac{\omega q}{\sqrt{2 E}}\right)$$

  Deduce that

  $$p=\frac{\partial S}{\partial q}=\frac{\partial W}{\partial q}=\sqrt{\left(2 E-\omega^{2} q^{2}\right)}$$

  from which

  $$p=\sqrt{2 E} \cos [\omega(t+\beta)] .$$

  Hence, or otherwise, show that the transformation from variables $(q, p)$ to $(\alpha, \beta)$ is canonical.

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• Paper 1, Section I, G

  #Coding and Cryptography

  2011

  I think of an integer $n$ with $1 \leqslant n \leqslant 10^{6}$. Explain how to find $n$ using twenty questions (or less) of the form 'Is it true that $n \geqslant m$ ?' to which I answer yes or no.

  I have watched a horse race with 15 horses. Is it possible to discover the order in which the horses finished by asking me twenty questions to which I answer yes or no?

  Roughly how many questions of the yes/no type are required to discover the order in which $n$ horses finished if $n$ is large?

  [You may assume that I answer honestly.]

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• Paper 2, Section I, G

  #Coding and Cryptography

  2011

  I happen to know that an apparently fair coin actually has probability $p$ of heads with $1>p>1 / 2$. I play a very long sequence of games of heads and tails in which my opponent pays me back twice my stake if the coin comes down heads and takes my stake if the coin comes down tails. I decide to bet a proportion $\alpha$ of my fortune at the end of the $n$th game in the $(n+1)$ st game. Determine, giving justification, the value $\alpha_{0}$ maximizing the expected logarithm of my fortune in the long term, assuming I use the same $\alpha_{0}$ at each game. Can it be actually disadvantageous for me to choose an $\alpha<\alpha_{0}$ (in the sense that I would be better off not playing)? Can it be actually disadvantageous for me to choose an $\alpha>\alpha_{0}$ ?

  [Moral issues should be ignored.]

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• Paper 3, Section I, G

  #Coding and Cryptography

  2011

  What is the rank of a binary linear code $C ?$ What is the weight enumeration polynomial $W_{C}$ of $C ?$

  Show that $W_{C}(1,1)=2^{r}$ where $r$ is the rank of $C$. Show that $W_{C}(s, t)=W_{C}(t, s)$ for all $s$ and $t$ if and only if $W_{C}(1,0)=1$.

  Find, with reasons, the weight enumeration polynomial of the repetition code of length $n$, and of the simple parity check code of length $n$.

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• Paper 4, Section I, G

  #Coding and Cryptography

  2011

  Describe a scheme for sending messages based on quantum theory which is not vulnerable to eavesdropping. You may ignore engineering problems.

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• Paper 1, Section II, $12 G$

  #Coding and Cryptography

  2011

  Describe the Rabin-Williams coding scheme. Show that any method for breaking it will enable us to factorise the product of two primes.

  Explain how the Rabin-Williams scheme can be used for bit sharing (that is to say 'tossing coins by phone').

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• Paper 2, Section II, G

  #Coding and Cryptography

  2011

  Define a cyclic code. Show that there is a bijection between the cyclic codes of length $n$ and the factors of $X^{n}-1$ over the field $\mathbb{F}_{2}$ of order 2 .

  What is meant by saying that $\alpha$ is a primitive $n$th root of unity in a finite field extension $K$ of $\mathbb{F}_{2}$ ? What is meant by saying that $C$ is a BCH code of length $n$ with defining set $\left\{\alpha, \alpha^{2}, \ldots, \alpha^{\delta-1}\right\}$ ? Show that such a code has minimum distance at least $\delta$.

  Suppose that $K$ is a finite field extension of $\mathbb{F}_{2}$ in which $X^{7}-1$ factorises into linear factors. Show that if $\beta$ is a root of $X^{3}+X^{2}+1$ then $\beta$ is a primitive 7 th root of unity and $\beta^{2}$ is also a root of $X^{3}+X^{2}+1$. Quoting any further results that you need show that the $\mathrm{BCH}$ code of length 7 with defining set $\left\{\beta, \beta^{2}\right\}$ is the Hamming code.

  [Results on the Vandermonde determinant may be used without proof provided they are quoted correctly.]

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• Paper 1, Section I, E

  #Cosmology

  2011

  Light of wavelength $\lambda_{e}$ emitted by a distant object is observed by us to have wavelength $\lambda_{0}$. The redshift $z$ of the object is defined by

  $$1+z=\frac{\lambda_{0}}{\lambda_{e}}$$

  Assuming that the object is at a fixed comoving distance from us in a homogeneous and isotropic universe with scale factor $a(t)$, show that

  $$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{e}\right)}$$

  where $t_{e}$ is the time of emission and $t_{0}$ the time of observation (i.e. today).

  [You may assume the non-relativistic Doppler shift formula $\Delta \lambda / \lambda=(v / c) \cos \theta$ for the shift $\Delta \lambda$ in the wavelength of light emitted by a nearby object travelling with velocity $v$ at angle $\theta$ to the line of sight.]

  Given that the object radiates energy $L$ per unit time, explain why the rate at which energy passes through a sphere centred on the object and intersecting the Earth is $L /(1+z)^{2}$.

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• Paper 2, Section I, E

  #Cosmology

  2011

  A spherically symmetric star in hydrostatic equilibrium has density $\rho(r)$ and pressure $P(r)$, which satisfy the pressure support equation,

  $$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$$

  where $m(r)$ is the mass within a radius $r$. Show that this implies

  $$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho .$$

  Provide a justification for choosing the boundary conditions $d P / d r=0$ at the centre of the $\operatorname{star}(r=0)$ and $P=0$ at its outer radius $(r=R)$.

  Use the pressure support equation $(*)$ to derive the virial theorem for a star,

  $$\langle P\rangle V=-\frac{1}{3} E_{\mathrm{grav}}$$

  where $\langle P\rangle$ is the average pressure, $V$ is the total volume of the star and $E_{\text {grav }}$ is its total gravitational potential energy.

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• Paper 3, Section I, E

  #Cosmology

  2011

  For an ideal gas of fermions of mass $m$ in volume $V$, and at temperature $T$ and chemical potential $\mu$, the number density $n$ and kinetic energy $E$ are given by

  $$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \bar{n}(p) p^{2} d p, \quad E=\frac{4 \pi g_{s}}{h^{3}} V \int_{0}^{\infty} \bar{n}(p) \epsilon(p) p^{2} d p$$

  where $g_{s}$ is the spin-degeneracy factor, $h$ is Planck's constant, $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the single-particle energy as a function of the momentum $p$, and

  $$\bar{n}(p)=\left[\exp \left(\frac{\epsilon(p)-\mu}{k T}\right)+1\right]^{-1}$$

  where $k$ is Boltzmann's constant.

  (i) Sketch the function $\bar{n}(p)$ at zero temperature, explaining why $\bar{n}(p)=0$ for $p>p_{F}$ (the Fermi momentum). Find an expression for $n$ at zero temperature as a function of $p F$.

  Assuming that a typical fermion is ultra-relativistic $\left(p c \gg m c^{2}\right)$ even at zero temperature, obtain an estimate of the energy density $E / V$ as a function of $p_{F}$, and hence show that

  $$E \sim h c n^{4 / 3} V$$

  in the ultra-relativistic limit at zero temperature.

  (ii) A white dwarf star of radius $R$ has total mass $M=\frac{4 \pi}{3} m_{p} n_{p} R^{3}$, where $m_{p}$ is the proton mass and $n_{p}$ the average proton number density. On the assumption that the star's degenerate electrons are ultra-relativistic, so that $(*)$ applies with $n$ replaced by the average electron number density $n_{e}$, deduce the following estimate for the star's internal kinetic energy:

  $$E_{\mathrm{kin}} \sim h c\left(\frac{M}{m_{p}}\right)^{4 / 3} \frac{1}{R} .$$

  By comparing this with the total gravitational potential energy, briefly discuss the consequences for white dwarf stability.

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• Paper 4, Section I, 10E

  #Cosmology

  2011

  The equilibrium number density of fermions at temperature $T$ is

  $$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{\infty} \frac{p^{2} d p}{\exp [(\epsilon(p)-\mu) / k T]+1}$$

  where $g_{s}$ is the spin degeneracy and $\epsilon(p)=c \sqrt{p^{2}+m^{2} c^{2}}$. For a non-relativistic gas with $p c \ll m c^{2}$ and $k T \ll m c^{2}-\mu$, show that the number density becomes

  $$n=g_{s}\left(\frac{2 \pi m k T}{h^{2}}\right)^{3 / 2} \exp \left[\left(\mu-m c^{2}\right) / k T\right]$$

  [You may assume that $\int_{0}^{\infty} d x x^{2} e^{-x^{2} / \alpha}=(\sqrt{\pi} / 4) \alpha^{3 / 2}$ for $\alpha>0$.]

  Before recombination, equilibrium is maintained between neutral hydrogen, free electrons, protons and photons through the interaction

  $$p+e^{-} \leftrightarrow H+\gamma$$

  Using the non-relativistic number density $(*)$, deduce Saha's equation relating the electron and hydrogen number densities,

  $$\frac{n_{e}^{2}}{n_{H}} \approx\left(\frac{2 \pi m_{e} k T}{h^{2}}\right)^{3 / 2} \exp (-I / k T)$$

  where $I=\left(m_{p}+m_{e}-m_{H}\right) c^{2}$ is the ionization energy of hydrogen. State clearly any assumptions you have made.

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• Paper 1, Section II, E

  #Cosmology

  2011

  A homogeneous and isotropic universe, with scale factor $a$, curvature parameter $k$, energy density $\rho$ and pressure $P$, satisfies the Friedmann and energy conservation equations

  $$\begin{aligned} &H^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho \\ &\dot{\rho}+3 H\left(\rho+P / c^{2}\right)=0 \end{aligned}$$

  where $H=\dot{a} / a$, and the dot indicates a derivative with respect to cosmological time $t$.

  (i) Derive the acceleration equation

  $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)$$

  Given that the strong energy condition $\rho c^{2}+3 P \geqslant 0$ is satisfied, show that $(a H)^{2}$ is a decreasing function of $t$ in an expanding universe. Show also that the density parameter $\Omega=8 \pi G \rho /\left(3 H^{2}\right)$ satisfies

  $$\Omega-1=\frac{k c^{2}}{a^{2} H^{2}}$$

  Hence explain, briefly, the flatness problem of standard big bang cosmology.

  (ii) A flat $(k=0)$ homogeneous and isotropic universe is filled with a radiation fluid $\left(w_{R}=1 / 3\right)$ and a dark energy fluid $\left(w_{\Lambda}=-1\right)$, each with an equation of state of the form $P_{i}=w_{i} \rho_{i} c^{2}$ and density parameters today equal to $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$ respectively. Given that each fluid independently obeys the energy conservation equation, show that the total energy density $\left(\rho_{R}+\rho_{\Lambda}\right) c^{2}$ equals $\rho c^{2}$, where

  $$\rho(t)=\frac{3 H_{0}^{2}}{8 \pi G} \frac{\Omega_{R 0}}{a^{4}}\left(1+\frac{1-\Omega_{R 0}}{\Omega_{R 0}} a^{4}\right)$$

  with $H_{0}$ being the value of the Hubble parameter today. Hence solve the Friedmann equation to get

  $$a(t)=\alpha(\sinh \beta t)^{1 / 2}$$

  where $\alpha$ and $\beta$ should be expressed in terms $\Omega_{R 0}$ and $\Omega_{\Lambda 0}$. Show that this result agrees with the expected asymptotic solutions at both early $(t \rightarrow 0)$ and late $(t \rightarrow \infty)$ times.

  [Hint: $\int d x / \sqrt{x^{2}+1}=\operatorname{arcsinh} x$.]

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• Paper 3, Section II, E

  #Cosmology

  2011

  An expanding universe with scale factor $a(t)$ is filled with (pressure-free) cold dark matter (CDM) of average mass density $\bar{\rho}(t)$. In the Zel'dovich approximation to gravitational clumping, the perturbed position $\mathbf{r}(\mathbf{q}, t)$ of a CDM particle with unperturbed comoving position $\mathbf{q}$ is given by

  $$\mathbf{r}(\mathbf{q}, t)=a(t)[\mathbf{q}+\boldsymbol{\psi}(\mathbf{q}, t)]$$

  where $\psi$ is the comoving displacement.

  (i) Explain why the conservation of CDM particles implies that

  $$\rho(\mathbf{r}, t) d^{3} r=a^{3} \bar{\rho}(t) d^{3} q,$$

  where $\rho(\mathbf{r}, t)$ is the CDM mass density. Use (1) to verify that $d^{3} q=a^{-3}\left[1-\nabla_{\mathbf{q}} \cdot \psi\right] d^{3} r$, and hence deduce that the fractional density perturbation is, to first order,

  $$\delta \equiv \frac{\rho-\bar{\rho}}{\bar{\rho}}=-\nabla_{\mathbf{q}} \cdot \psi \text {. }$$

  Use this result to integrate the Poisson equation $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ for the gravitational potential $\Phi$. Then use the particle equation of motion $\ddot{\mathbf{r}}=-\nabla \Phi$ to deduce a second-order differential equation for $\psi$, and hence that

  $$\ddot{\delta}+2\left(\frac{\dot{a}}{a}\right) \dot{\delta}-4 \pi G \bar{\rho} \delta=0 .$$

  [You may assume that $\nabla^{2} \Phi=4 \pi G \bar{\rho}$ implies $\nabla \Phi=(4 \pi G / 3) \bar{\rho} \mathbf{r}$ and that the pressure-free acceleration equation is $\ddot{a}=-(4 \pi G / 3) \bar{\rho} a .]$

  (ii) A flat matter-dominated universe with background density $\bar{\rho}=\left(6 \pi G t^{2}\right)^{-1}$ has scale factor $a(t)=\left(t / t_{0}\right)^{2 / 3}$. The universe is filled with a pressure-free homogeneous (non-clumping) fluid of mass density $\rho_{H}(t)$, as well as cold dark matter of mass density $\rho_{C}(\mathbf{r}, t)$.

  Assuming that the Zel'dovich perturbation equation in this case is as in (2) but with $\bar{\rho}$ replaced by $\bar{\rho}_{C}$, i.e. that

  $$\ddot{\delta}+2\left(\frac{\dot{a}}{a}\right) \dot{\delta}-4 \pi G \bar{\rho}_{C} \delta=0,$$

  seek power-law solutions $\delta \propto t^{\alpha}$ to find growing and decaying modes with

  $$\alpha=\frac{1}{6}\left(-1 \pm \sqrt{25-24 \Omega_{H}}\right)$$

  where $\Omega_{H}=\rho_{H} / \bar{\rho}$.

  Given that matter domination starts $\left(t=t_{\text {eq }}\right)$ at a redshift $z \approx 10^{5}$, and given an initial perturbation $\delta\left(t_{\mathrm{eq}}\right) \approx 10^{-5}$, show that $\Omega_{H}=2 / 3$ yields a model that is not compatible with the large-scale structure observed today.

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• Paper 1, Section II, I

  #Differential Geometry

  2011

  Let $X$ and $Y$ be manifolds and $f: X \rightarrow Y$ a smooth map. Define the notions critical point, critical value, regular value of $f$. Prove that if $y$ is a regular value of $f$, then $f^{-1}(y)$ (if non-empty) is a smooth manifold of $\operatorname{dimension} \operatorname{dim} X-\operatorname{dim} Y$.

  [The Inverse Function Theorem may be assumed without proof if accurately stated.]

  Let $M_{n}(\mathbb{R})$ be the set of all real $n \times n$ matrices and $\operatorname{SO}(n) \subset M_{n}(\mathbb{R})$ the group of all orthogonal matrices with determinant 1 . Show that $\mathrm{SO}(n)$ is a smooth manifold and find its dimension.

  Show further that $\mathrm{SO}(n)$ is compact and that its tangent space at $A \in \operatorname{SO}(n)$ is given by all matrices $H$ such that $A H^{t}+H A^{t}=0$.

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• Paper 2, Section II, I

  #Differential Geometry

  2011

  Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a smooth curve parametrized by arc-length, with $\alpha^{\prime \prime}(s) \neq 0$ for all $s \in I$. Define what is meant by the Frenet frame $t(s), n(s), b(s)$, the curvature and torsion of $\alpha$. State and prove the Frenet formulae.

  By considering $\langle\alpha, t \times n\rangle$, or otherwise, show that, if for each $s \in I$ the vectors $\alpha(s)$, $t(s)$ and $n(s)$ are linearly dependent, then $\alpha(s)$ is a plane curve.

  State and prove the isoperimetric inequality for $C^{1}$ regular plane curves.

  [You may assume Wirtinger's inequality, provided you state it accurately.]

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• Paper 3, Section II, I

  #Differential Geometry

  2011

  For an oriented surface $S$ in $\mathbb{R}^{3}$, define the Gauss map, the second fundamental form and the normal curvature in the direction $w \in T_{p} S$ at a point $p \in S$.

  Let $\tilde{k}_{1}, \ldots, \tilde{k}_{m}$ be normal curvatures at $p$ in the directions $v_{1}, \ldots, v_{m}$, such that the angle between $v_{i}$ and $v_{i+1}$ is $\pi / m$ for each $i=1, \ldots, m-1(m \geqslant 2)$. Show that

  $$\tilde{k}_{1}+\ldots+\tilde{k}_{m}=m H$$

  where $H$ is the mean curvature of $S$ at $p$.

  What is a minimal surface? Show that if $S$ is a minimal surface, then its Gauss $\operatorname{map} N$ at each point $p \in S$ satisfies

  $$\left\langle d N_{p}\left(w_{1}\right), d N_{p}\left(w_{2}\right)\right\rangle=\mu(p)\left\langle w_{1}, w_{2}\right\rangle, \quad \text { for all } w_{1}, w_{2} \in T_{p} S,$$

  where $\mu(p) \in \mathbb{R}$ depends only on $p$. Conversely, if the identity $(*)$ holds at each point in $S$, must $S$ be minimal? Justify your answer.

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• Paper 4, Section II, I

  #Differential Geometry

  2011

  Define what is meant by a geodesic. Let $S \subset \mathbb{R}^{3}$ be an oriented surface. Define the geodesic curvature $k_{g}$ of a smooth curve $\gamma: I \rightarrow S$ parametrized by arc-length.

  Explain without detailed proofs what are the exponential map $\exp _{p}$ and the geodesic polar coordinates $(r, \theta)$ at $p \in S$. Determine the derivative $d\left(\exp _{p}\right)_{0}$. Prove that the coefficients of the first fundamental form of $S$ in the geodesic polar coordinates satisfy

  $$E=1, \quad F=0, \quad G(0, \theta)=0, \quad(\sqrt{G})_{r}(0, \theta)=1$$

  State the global Gauss-Bonnet formula for compact surfaces with boundary. [You should identify all terms in the formula.]

  Suppose that $S$ is homeomorphic to a cylinder $S^{1} \times \mathbb{R}$ and has negative Gaussian curvature at each point. Prove that $S$ has at most one simple (i.e. without selfintersections) closed geodesic.

  [Basic properties of geodesics may be assumed, if accurately stated.]

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• Paper 1, Section I, C

  #Dynamical Systems

  2011

  Find the fixed points of the dynamical system (with $\mu \neq 0$ )

  $$\begin{aligned} &\dot{x}=\mu^{2} x-x y \\ &\dot{y}=-y+x^{2} \end{aligned}$$

  and determine their type as a function of $\mu$.

  Find the stable and unstable manifolds of the origin correct to order $4 .$

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• Paper 2, Section I, C

  #Dynamical Systems

  2011

  State the Poincaré-Bendixson theorem for two-dimensional dynamical systems.

  A dynamical system can be written in polar coordinates $(r, \theta)$ as

  $$\begin{aligned} &\dot{r}=r-r^{3}(1+\alpha \cos \theta) \\ &\dot{\theta}=1-r^{2} \beta \cos \theta \end{aligned}$$

  where $\alpha$ and $\beta$ are constants with $0<\alpha<1$.

  Show that trajectories enter the annulus $(1+\alpha)^{-1 / 2}<r<(1-\alpha)^{-1 / 2}$.

  Show that if there is a fixed point $\left(r_{0}, \theta_{0}\right)$ inside the annulus then $r_{0}^{2}=(\beta-\alpha) / \beta$ and $\cos \theta_{0}=1 /(\beta-\alpha)$.

  Use the Poincaré-Bendixson theorem to derive conditions on $\beta$ that guarantee the existence of a periodic orbit.

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• Paper 3, Section I, C

  #Dynamical Systems

  2011

  For the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$, with $\lambda>0$, show the following:

  (i) If $\lambda<1$, then the origin is the only fixed point and is stable.

  (ii) If $\lambda>1$, then the origin is unstable. There are two further fixed points which are stable for $1<\lambda<2$ and unstable for $\lambda>2$.

  (iii) If $\lambda<3 \sqrt{3} / 2$, then $x_{n}$ has the same sign as the starting value $x_{0}$ if $\left|x_{0}\right|<1$.

  (iv) If $\lambda<3$, then $\left|x_{n+1}\right|<2 \sqrt{3} / 3$ when $\left|x_{n}\right|<2 \sqrt{3} / 3$. Deduce that iterates starting sufficiently close to the origin remain bounded, though they may change sign.

  [Hint: For (iii) and (iv) a graphical representation may be helpful.]

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• Paper 4, Section I, $7 \mathrm{C}$

  #Dynamical Systems

  2011

  (i) Explain the use of the energy balance method for describing approximately the behaviour of nearly Hamiltonian systems.

  (ii) Consider the nearly Hamiltonian dynamical system

  $$\ddot{x}+\epsilon \dot{x}\left(-1+\alpha x^{2}-\beta x^{4}\right)+x=0, \quad 0<\epsilon \ll 1$$

  where $\alpha$ and $\beta$ are positive constants. Show that, for sufficiently small $\epsilon$, the system has periodic orbits if $\alpha^{2}>8 \beta$, and no periodic orbits if $\alpha^{2}<8 \beta$. Show that in the first case there are two periodic orbits, and determine their approximate size and their stability.

  What can you say about the existence of periodic orbits when $\alpha^{2}=8 \beta ?$

  [You may assume that

  $$\left.\int_{0}^{2 \pi} \sin ^{2} t d t=\pi, \quad \int_{0}^{2 \pi} \sin ^{2} t \cos ^{2} t d t=\frac{\pi}{4}, \quad \int_{0}^{2 \pi} \sin ^{2} t \cos ^{4} t d t=\frac{\pi}{8}\right]$$

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• Paper 3, Section II, C

  #Dynamical Systems

  2011

  Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$ in $\mathbb{R}^{n}$, where $\mu$ is a real parameter.

  Consider the system in $x \geqslant 0, y \geqslant 0$, with $\mu>0$,

  $$\begin{aligned} \dot{x} &=x\left(1-y^{2}-x^{2}\right), \\ \dot{y} &=y\left(\mu-y-x^{2}\right) . \end{aligned}$$

  (i) Show that both the fixed point $(0, \mu)$ and the fixed point $(1,0)$ have a steady-state bifurcation when $\mu=1$.

  (ii) By finding the first approximation to the extended centre manifold, construct the normal form near the bifurcation point $(1,0)$ when $\mu$ is close to unity, and show that there is a transcritical bifurcation there. Explain why the symmetries of the equations mean that the bifurcation at $(0,1)$ must be of pitchfork type.

  (iii) Show that two fixed points with $x, y>0$ exist in the range $1<\mu<5 / 4$. Show that the solution with $y<1 / 2$ is stable. Identify the bifurcation that occurs at $\mu=5 / 4$.

  (iv) Draw a sketch of the values of $y$ at the fixed points as functions of $\mu$, indicating the bifurcation points and the regions where each branch is stable. [Detailed calculations are not required.]

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• Paper 4, Section II, C

  #Dynamical Systems

  2011

  (i) State and prove Lyapunov's First Theorem, and state (without proof) La Salle's Invariance Principle. Show by example how the latter result can be used to prove asymptotic stability of a fixed point even when a strict Lyapunov function does not exist.

  (ii) Consider the system

  $$\begin{aligned} &\dot{x}=-x+2 y+x^{3}+2 x^{2} y+2 x y^{2}+2 y^{3}, \\ &\dot{y}=-y-x-2 x^{3}+\frac{1}{2} x^{2} y-3 x y^{2}+y^{3} \end{aligned}$$

  Show that the origin is asymptotically stable and that the basin of attraction of the origin includes the region $x^{2}+2 y^{2}<2 / 3$.

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• Paper 1, Section II, C

  #Electrodynamics

  2011

  In the Landau-Ginzburg model of superconductivity, the energy of the system is given, for constants $\alpha$ and $\beta$, by

  $$E=\int\left\{\frac{1}{2 \mu_{0}} \mathbf{B}^{2}+\frac{1}{2 m}\left[(i \hbar \nabla-q \mathbf{A}) \psi^{*} \cdot(-i \hbar \nabla-q \mathbf{A}) \psi\right]+\alpha \psi^{*} \psi+\beta\left(\psi^{*} \psi\right)^{2}\right\} d^{3} \mathbf{x},$$

  where $\mathbf{B}$ is the time-independent magnetic field derived from the vector potential $\mathbf{A}$, and $\psi$ is the wavefunction of the charge carriers, which have mass $m$ and charge $q$.

  Describe the physical meaning of each of the terms in the integral.

  Explain why in a superconductor one must choose $\alpha<0$ and $\beta>0$. Find an expression for the number density $n$ of the charge carriers in terms of $\alpha$ and $\beta$.

  Show that the energy is invariant under the gauge transformations

  $$\mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda, \quad \psi \rightarrow \psi e^{i q \Lambda / \hbar}$$

  Assuming that the number density $n$ is uniform, show that, if $E$ is a minimum under variations of $\mathbf{A}$, then

  $$\operatorname{curl} \mathbf{B}=-\frac{\mu_{0} q^{2} n}{m}\left(\mathbf{A}-\frac{\hbar}{q} \nabla \phi\right)$$

  where $\phi=\arg \psi$.

  Find a formula for $\nabla^{2} \mathbf{B}$ and use it to explain why there cannot be a magnetic field inside the bulk of a superconductor.

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• Paper 3, Section II, C

  #Electrodynamics

  2011

  Explain how time-dependent distributions of electric charge $\rho(\mathbf{x}, t)$ and current $\mathbf{j}(\mathbf{x}, t)$ can be combined into a four-vector $j^{a}(x)$ that obeys $\partial_{a} j^{a}=0$.

  This current generates a four-vector potential $A^{a}(x)$. Explain how to find $A^{a}$ in the gauge $\partial_{a} A^{a}=0$.

  A small circular loop of wire of radius $r$ is centred at the origin. The unit vector normal to the plane of the loop is $\mathbf{n}$. A current $I_{o} \sin \omega t$ flows in the loop. Find the three-vector potential $\mathbf{A}(\mathbf{x}, t)$ to leading order in $r /|\mathbf{x}|$.

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• Paper 4, Section II, C

  #Electrodynamics

  2011

  Suppose that there is a distribution of electric charge given by the charge density $\rho(\mathbf{x})$. Develop the multipole expansion, up to quadrupole terms, for the electrostatic potential $\phi$ and define the dipole and quadrupole moments of the charge distribution.

  A tetrahedron has a vertex at $(1,1,1)$ where there is a point charge of strength $3 q$. At each of the other vertices located at $(1,-1,-1),(-1,1,-1)$ and $(-1,-1,1)$ there is a point charge of strength $-q$.

  What is the dipole moment of this charge distribution?

  What is the quadrupole moment?

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• Paper 1, Section II, B

  #Fluid Dynamics II

  2011

  The steady two-dimensional boundary-layer equations for flow primarily in the $x$ direction are

  $$\begin{gathered} \rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=-\frac{d P}{d x}+\mu \frac{\partial^{2} u}{\partial y^{2}} \\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \end{gathered}$$

  A thin, steady, two-dimensional jet emerges from a point at the origin and flows along the $x$-axis in a fluid at rest far from the $x$-axis. Show that the momentum flux

  $$F=\int_{-\infty}^{\infty} \rho u^{2} d y$$

  is independent of position $x$ along the jet. Deduce that the thickness $\delta(x)$ of the jet increases along the jet as $x^{2 / 3}$, while the centre-line velocity $U(x)$ decreases as $x^{-1 / 3}$.

  A similarity solution for the jet is sought with a streamfunction $\psi$ of the form

  $$\psi(x, y)=U(x) \delta(x) f(\eta) \quad \text { with } \quad \eta=y / \delta(x) .$$

  Derive the nonlinear third-order non-dimensional differential equation governing $f$, and write down the boundary and normalisation conditions which must be applied.

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• Paper 2, Section II, B

  #Fluid Dynamics II

  2011

  The energy equation for the motion of a viscous, incompressible fluid states that

  $$\frac{d}{d t} \int_{V} \frac{1}{2} \rho u^{2} d V+\int_{S} \frac{1}{2} \rho u^{2} u_{i} n_{i} d S=\int_{S} u_{i} \sigma_{i j} n_{j} d S-2 \mu \int_{V} e_{i j} e_{i j} d V$$

  Interpret each term in this equation and explain the meaning of the symbols used.

  Consider steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls. Deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient $G$, and the volume flux $Q$.

  Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius $a$. Using the relationship derived above, or otherwise, find the viscous dissipation per unit length of this flow in terms of $G$.

  [Hint: In cylindrical polar coordinates,

  $$\nabla^{2} w(r)=\frac{1}{r} \frac{d}{d r}\left(r \frac{d w}{d r}\right)$$

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• Paper 3, Section II, B

  #Fluid Dynamics II

  2011

  If $A_{i}\left(x_{j}\right)$ is harmonic, i.e. if $\nabla^{2} A_{i}=0$, show that

  $$u_{i}=A_{i}-x_{k} \frac{\partial A_{k}}{\partial x_{i}}, \quad \text { with } \quad p=-2 \mu \frac{\partial A_{n}}{\partial x_{n}},$$

  satisfies the incompressibility condition and the Stokes equation. Show that the stress tensor is

  $$\sigma_{i j}=2 \mu\left(\delta_{i j} \frac{\partial A_{n}}{\partial x_{n}}-x_{k} \frac{\partial^{2} A_{k}}{\partial x_{i} \partial x_{j}}\right)$$

  Consider the Stokes flow corresponding to

  $$A_{i}=V_{i}\left(1-\frac{a}{2 r}\right),$$

  where $V_{i}$ are the components of a constant vector $\mathbf{V}$. Show that on the sphere $r=a$ the normal component of velocity vanishes and the surface traction $\sigma_{i j} x_{j} / a$ is in the normal direction. Hence deduce that the drag force on the sphere is given by

  $$\mathbf{F}=4 \pi \mu a \mathbf{V}$$

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• Paper 4, Section II, B

  #Fluid Dynamics II

  2011

  A viscous fluid flows along a slowly varying thin channel between no-slip surfaces at $y=0$ and $y=h(x, t)$ under the action of a pressure gradient $d p / d x$. After explaining the approximations and assumptions of lubrication theory, including a comment on the reduced Reynolds number, derive the expression for the volume flux

  $$q=\int_{0}^{h} u d y=-\frac{h^{3}}{12 \mu} \frac{d p}{d x}$$

  as well as the equation

  $$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$$

  In peristaltic pumping, the surface $h(x, t)$ has a periodic form in space which propagates at a constant speed $c$, i.e. $h(x-c t)$, and no net pressure gradient is applied, i.e. the pressure gradient averaged over a period vanishes. Show that the average flux along the channel is given by

  $$\langle q\rangle=c\left(\langle h\rangle-\frac{\left\langle h^{-2}\right\rangle}{\left\langle h^{-3}\right\rangle}\right)$$

  where $\langle\cdot\rangle$ denotes an average over one period.

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• Paper 1, Section I, E

  #Further Complex Methods

  2011

  Show that the following integral is well defined:

  $$I(a, b)=\int_{0}^{\infty}\left(\frac{e^{-b x}}{e^{i a} e^{x}-1}-\frac{e^{b x}}{e^{-i a} e^{x}-1}\right) d x, \quad 0<a<\infty, a \neq 2 n \pi, n \in \mathbb{Z}, 0<b<1$$

  Express $I(a, b)$ in terms of a combination of hypergeometric functions.

  [You may assume without proof that the hypergeometric function $F(a, b ; c ; z)$ can be expressed in the form

  $$F(a, b ; c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

  for appropriate restrictions on $c, b, z$. Furthermore,

  $$\Gamma(z+1)=z \Gamma(z) \cdot]$$

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• Paper 2, Section I, E

  #Further Complex Methods

  2011

  Find the two complex-valued functions $F^{+}(z)$ and $F^{-}(z)$ such that all of the following hold:

  (i) $F^{+}(z)$ and $F^{-}(z)$ are analytic for $\operatorname{Im} z>0$ and $\operatorname{Im} z<0$ respectively, where $z=x+i y, x, y \in \mathbb{R}$.

  (ii) $F^{+}(x)-F^{-}(x)=\frac{1}{x^{4}+1}, \quad x \in \mathbb{R}$.

  (iii) $F^{\pm}(z)=O\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0$.

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• Paper 3, Section I, E

  #Further Complex Methods

  2011

  Explain the meaning of $z_{j}$ in the Weierstrass canonical product formula

  $$f(z)=f(0) \exp \left[\frac{f^{\prime}(0)}{f(0)} z\right] \prod_{j=1}^{\infty}\left\{\left(1-\frac{z}{z_{j}}\right) e^{\frac{z}{z_{j}}}\right\}$$

  Show that

  $$\frac{\sin (\pi z)}{\pi z}=\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)$$

  Deduce that

  $$\pi \cot (\pi z)=\frac{1}{z}+2 \sum_{n=1}^{\infty} \frac{z}{z^{2}-n^{2}}$$

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• Paper 4, Section $I$, E

  #Further Complex Methods

  2011

  Let $F(z)$ be defined by

  $$F(z)=\int_{0}^{\infty} \frac{e^{-z t}}{1+t^{2}} d t, \quad|\arg z|<\frac{\pi}{2}$$

  Let $\tilde{F}(z)$ be defined by

  $$\tilde{F}(z)=\mathcal{P} \int_{0}^{\infty e^{-\frac{i \pi}{2}}} \frac{e^{-z \zeta}}{1+\zeta^{2}} d \zeta, \quad 0<\arg z<\pi$$

  where $\mathcal{P}$ denotes principal value integral and the contour is the negative imaginary axis.

  By computing $F(z)-\tilde{F}(z)$, obtain a formula for the analytic continuation of $F(z)$ for $\frac{\pi}{2} \leqslant \arg z<\pi$.

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• Paper 1, Section II, E

  #Further Complex Methods

  2011

  (i) By assuming the validity of the Fourier transform pair, prove the validity of the following transform pair:

  $$\begin{gathered} \hat{q}(k)=\int_{0}^{\infty} e^{-i k x} q(x) d x, \quad \operatorname{Im} k \leqslant 0, \\ q(x)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i k x} \hat{q}(k) d k+\frac{c}{2 \pi} \int_{L} e^{i k x} \hat{q}(-k) d k, \quad 0<x<\infty, \end{gathered}$$

  where $c$ is an arbitrary complex constant and $L$ is the union of the two rays arg $k=\frac{\pi}{2}$ and $\arg k=0$ with the orientation shown in the figure below:

  

  The contour $L$.

  (ii) Verify that the partial differential equation

  $$i q_{t}+q_{x x}=0, \quad 0<x<\infty, t>0,$$

  can be rewritten in the following form:

  $$\left(e^{-i k x+i k^{2} t} q\right)_{t}-\left[e^{-i k x+i k^{2} t}\left(-k q+i q_{x}\right)\right]_{x}=0, \quad k \in \mathbb{C} .$$

  Consider equation (2) supplemented with the conditions

  $$\begin{aligned} &q(x, 0)=q_{0}(x), \quad 0<x<\infty \\ &q(x, t) \text { vanishes sufficiently fast for all } t \text { as } x \rightarrow \infty \end{aligned}$$

  By using equations (1a) and (3), show that

  $$\hat{q}(k, t)=e^{-i k^{2} t} \hat{q}_{0}(k)+e^{-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)-i \tilde{g}_{1}\left(k^{2}, t\right)\right], \operatorname{Im} k \leqslant 0$$

  where

  $$\hat{q}_{0}(k)=\int_{0}^{\infty} e^{-i k x} q_{0}(x) d x, \quad \operatorname{Im} k \leqslant 0$$

  Part II, $2011 \quad$ List of Questions

  [TURN OVER

  $$\tilde{g}_{0}(k, t)=\int_{0}^{t} e^{i k \tau} q(0, \tau) d \tau, \quad \tilde{g}_{1}(k, t)=\int_{0}^{t} e^{i k \tau} q_{x}(0, \tau) d \tau, k \in \mathbb{C}, t>0 .$$

  Use (1b) to invert equation (5) and furthermore show that

  $$\int_{-\infty}^{\infty} e^{i k x-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)+i \tilde{g}_{1}\left(k^{2}, t\right)\right] d k=\int_{L} e^{i k x-i k^{2} t}\left[k \tilde{g}_{0}\left(k^{2}, t\right)+i \tilde{g}_{1}\left(k^{2}, t\right)\right] d k, t>0, x>0$$

  Hence determine the constant $c$ so that the solution of equation (2), with the conditions (4) and with the condition that either $q(0, t)$ or $q_{x}(0, t)$ is given, can be expressed in terms of an integral involving $\hat{q}_{0}(k)$ and either $\tilde{g}_{0}$ or $\tilde{g}_{1}$.

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• Paper 2, Section II, E

  #Further Complex Methods

  2011

  Consider the following sum related to Riemann's zeta function:

  $$S:=\sum_{m=1}^{\left[\frac{a}{2 \pi}\right]} m^{s-1}, \quad s=\sigma+i t, \sigma, t \in \mathbb{R}, \quad a>2 \pi, a \neq 2 \pi N, N \in \mathbb{Z}^{+},$$

  where $[a / 2 \pi]$ denotes the integer part of $a / 2 \pi$.

  (i) By using an appropriate branch cut, show that

  $$S=\frac{e^{-\frac{i \pi s}{2}}}{(2 \pi)^{s}} \int_{L} f(z, s) d z, \quad f(z, s)=\frac{e^{-z}}{1-e^{-z}} z^{s-1}$$

  where $L$ is the circle in the complex $z$-plane centred at $i(a+b) / 2$ with radius $(a-b) / 2$, $0<b<2 \pi$.

  (ii) Use the above representation to show that, for $a>2 \pi$ and $0<b<2 \pi$,

  $$\sum_{m=1}^{\left[\frac{a}{2 \pi}\right]} m^{s-1}=\frac{1}{(2 \pi)^{s}}\left[e^{-\frac{i \pi s}{2}} \int_{C_{b}^{a}} f(z, s) d z-e^{\frac{i \pi s}{2}} \int_{C_{-a}^{-b}} f(z, s) d z+\frac{a^{s}}{s}-\frac{b^{s}}{s}\right]$$

  where $f(z, s)$ is defined in (i) and the curves $C_{b}^{a}, C_{-a}^{-b}$ are the following semi-circles in the right half complex $z$-plane:

  

  The curves $C_{b}^{a}$ and $C_{-a}^{-b}$.

  $$\begin{aligned} C_{b}^{a} &=\left\{\frac{i(a+b)}{2}+\frac{(a-b)}{2} e^{i \theta}, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\right\} \\ C_{-a}^{-b} &=\left\{\frac{-i(a+b)}{2}+\frac{(a-b)}{2} e^{i \theta}, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\right\} \end{aligned}$$

  Part II, 2011 List of Questions

  [TURN OVER

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• Paper 1, Section II, 18H

  #Galois Theory

  2011

  Let $K$ be a field.

  (i) Let $F$ and $F^{\prime}$ be two finite extensions of $K$. When the degrees of these two extensions are equal, show that every $K$-homomorphism $F \rightarrow F^{\prime}$ is an isomorphism. Give an example, with justification, of two finite extensions $F$ and $F^{\prime}$ of $K$, which have the same degrees but are not isomorphic over $K$.

  (ii) Let $L$ be a finite extension of $K$. Let $F$ and $F^{\prime}$ be two finite extensions of $L$. Show that if $F$ and $F^{\prime}$ are isomorphic as extensions of $L$ then they are isomorphic as extensions of $K$. Prove or disprove the converse.

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• Paper 2, Section II, H

  #Galois Theory

  2011

  Let $F=\mathbb{C}(x, y)$ be the function field in two variables $x, y$. Let $n \geqslant 1$, and $K=\mathbb{C}\left(x^{n}+y^{n}, x y\right)$ be the subfield of $F$ of all rational functions in $x^{n}+y^{n}$ and $x y .$

  (i) Let $K^{\prime}=K\left(x^{n}\right)$, which is a subfield of $F$. Show that $K^{\prime} / K$ is a quadratic extension.

  (ii) Show that $F / K^{\prime}$ is cyclic of order $n$, and $F / K$ is Galois. Determine the Galois $\operatorname{group} \operatorname{Gal}(F / K)$.

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• Paper 3, Section II, H

  #Galois Theory

  2011

  Let $n \geqslant 1$ and $K=\mathbb{Q}\left(\boldsymbol{\mu}_{n}\right)$ be the cyclotomic field generated by the $n$th roots of unity. Let $a \in \mathbb{Q}$ with $a \neq 0$, and consider $F=K(\sqrt[n]{a})$.

  (i) State, without proof, the theorem which determines $\operatorname{Gal}(K / \mathbb{Q})$.

  (ii) Show that $F / \mathbb{Q}$ is a Galois extension and that $\operatorname{Gal}(F / \mathbb{Q})$ is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]

  (iii) When $n=p$ is prime, list all possible degrees $[F: \mathbb{Q}]$, with justification.

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• Paper 4, Section II, H

  #Galois Theory

  2011

  Let $K$ be a field of characteristic 0 , and let $P(X)=X^{4}+b X^{2}+c X+d$ be an irreducible quartic polynomial over $K$. Let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ be its roots in an algebraic closure of $K$, and consider the Galois group $\operatorname{Gal}(P)$ (the group $\operatorname{Gal}(F / K)$ for a splitting field $F$ of $P$ over $K$ ) as a subgroup of $S_{4}$ (the group of permutations of $\left.\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right)$.

  Suppose that $\operatorname{Gal}(P)$ contains $V_{4}=\{1,(12)(34),(13)(24),(14)(23)\}$.

  (i) List all possible $\operatorname{Gal}(P)$ up to isomorphism. [Hint: there are 4 cases, with orders 4 , 8,12 and 24.]

  (ii) Let $Q(X)$ be the resolvent cubic of $P$, i.e. a cubic in $K[X]$ whose roots are $-\left(\alpha_{1}+\alpha_{2}\right)\left(\alpha_{3}+\alpha_{4}\right),-\left(\alpha_{1}+\alpha_{3}\right)\left(\alpha_{2}+\alpha_{4}\right)$ and $-\left(\alpha_{1}+\alpha_{4}\right)\left(\alpha_{2}+\alpha_{3}\right)$. Construct a natural surjection $\operatorname{Gal}(P) \rightarrow \operatorname{Gal}(Q)$, and find $\operatorname{Gal}(Q)$ in each of the four cases found in (i).

  (iii) Let $\Delta \in K$ be the discriminant of $Q$. Give a criterion to determine $\operatorname{Gal}(P)$ in terms of $\Delta$ and the factorisation of $Q$ in $K[X]$.

  (iv) Give a specific example of $P$ where $\operatorname{Gal}(P)$ is abelian.

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• Paper 1, Section II, 37D

  #General Relativity

  2011

  Consider a metric of the form

  $$d s^{2}=-2 d u d v+d x^{2}+d y^{2}-2 H(u, x, y) d u^{2} .$$

  Let $x^{a}(\lambda)$ describe an affinely-parametrised geodesic, where $x^{a} \equiv\left(x^{1}, x^{2}, x^{3}, x^{4}\right)=$ $(u, v, x, y)$. Write down explicitly the Lagrangian

  $$L=g_{a b} \dot{x}^{a} \dot{x}^{b},$$

  with $\dot{x}^{a}=d x^{a} / d \lambda$, using the given metric. Hence derive the four geodesic equations. In particular, show that

  $$\ddot{v}+2\left(\frac{\partial H}{\partial x} \dot{x}+\frac{\partial H}{\partial y} \dot{y}\right) \dot{u}+\frac{\partial H}{\partial u} \dot{u}^{2}=0$$

  By comparing these equations with the standard form of the geodesic equation, show that $\Gamma_{13}^{2}=\partial H / \partial x$ and derive the other Christoffel symbols.

  The Ricci tensor, $R_{a b}$, is defined by

  $$R_{a b}=\Gamma_{a b, d}^{d}-\Gamma_{a d, b}^{d}+\Gamma_{d f}^{d} \Gamma_{b a}^{f}-\Gamma_{b f}^{d} \Gamma_{d a}^{f}$$

  By considering the case $a=1, b=1$, show that the vacuum Einstein field equations imply

  $$\frac{\partial^{2} H}{\partial x^{2}}+\frac{\partial^{2} H}{\partial y^{2}}=0$$

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• Paper 2, Section II, 36D

  #General Relativity

  2011

  The curvature tensor $R_{b c d}^{a}$ satisfies

  $$V_{a ; b c}-V_{a ; c b}=V_{e} R_{a b c}^{e}$$

  for any covariant vector field $V_{a}$. Hence express $R_{a b c}^{e}$ in terms of the Christoffel symbols and their derivatives. Show that

  $$R_{a b c}^{e}=-R_{a c b}^{e}$$

  Further, by setting $V_{a}=\partial \phi / \partial x^{a}$, deduce that

  $$R_{a b c}^{e}+R_{c a b}^{e}+R_{b c a}^{e}=0 .$$

  Using local inertial coordinates or otherwise, obtain the Bianchi identities.

  Define the Ricci tensor in terms of the curvature tensor and show that it is symmetric. [You may assume that $R_{a b c d}=-R_{b a c d}$.] Write down the contracted Bianchi identities.

  In certain spacetimes of dimension $n \geqslant 2, R_{a b c d}$ takes the form

  $$R_{a b c d}=K\left(g_{a c} g_{b d}-g_{a d} g_{b c}\right)$$

  Obtain the Ricci tensor and curvature scalar. Deduce, under some restriction on $n$ which should be stated, that $K$ is a constant.

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• Paper 4, Section II, D

  #General Relativity

  2011

  The metric of the Schwarzschild solution is

  $$d s^{2}=-\left(1-\frac{2 M}{r}\right) d t^{2}+\frac{1}{\left(1-\frac{2 M}{r}\right)} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right) .$$

  Show that, for an incoming radial light ray, the quantity

  $$v=t+r+2 M \log \left|\frac{r}{2 M}-1\right|$$

  is constant.

  Express $d s^{2}$ in terms of $r, v, \theta$ and $\phi$. Determine the light-cone structure in these coordinates, and use this to discuss the nature of the apparent singularity at $r=2 M$.

  An observer is falling radially inwards in the region $r<2 M$. Assuming that the metric for $r<2 M$ is again given by $(*)$, obtain a bound for $d \tau$, where $\tau$ is the proper time of the observer, in terms of $d r$. Hence, or otherwise, determine the maximum proper time that can elapse between the events at which the observer crosses $r=2 M$ and is torn apart at $r=0$.

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• Paper 1, Section I, G

  #Geometry and Groups

  2011

  Let $G$ be a finite subgroup of $\mathrm{SO}(3)$ and let $\Omega$ be the set of unit vectors that are fixed by some non-identity element of $G$. Show that the group $G$ permutes the unit vectors in $\Omega$ and that $\Omega$ has at most three orbits. Describe these orbits when $G$ is the group of orientation-preserving symmetries of a regular dodecahedron.

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• Paper 2, Section $I$, $3 \mathrm{G}$

  #Geometry and Groups

  2011

  Let $A$ and $B$ be two rotations of the Euclidean plane $\mathbb{E}^{2}$ about centres $a$ and $b$ respectively. Show that the conjugate $A B A^{-1}$ is also a rotation and find its fixed point. When do $A$ and $B$ commute? Show that the commutator $A B A^{-1} B^{-1}$ is a translation.

  Deduce that any group of orientation-preserving isometries of the Euclidean plane either fixes a point or is infinite.

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• Paper 3, Section I, $3 G$

  #Geometry and Groups

  2011

  Define a Kleinian group.

  Give an example of a Kleinian group that is a free group on two generators and explain why it has this property.

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• Paper 4, Section I, G

  #Geometry and Groups

  2011

  Define inversion in a circle $\Gamma$ on the Riemann sphere. You should show from your definition that inversion in $\Gamma$ exists and is unique.

  Prove that the composition of an even number of inversions is a Möbius transformation of the Riemann sphere and that every Möbius transformation is the composition of an even number of inversions.

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• Paper 1, Section II, G

  #Geometry and Groups

  2011

  Prove that a group of Möbius transformations is discrete if, and only if, it acts discontinuously on hyperbolic 3 -space.

  Let $G$ be the set of Möbius transformations $z \mapsto \frac{a z+b}{c z+d}$ with

  $$a, b, c, d \in \mathbb{Z}[i]=\{u+i v: u, v \in \mathbb{Z}\} \quad \text { and } \quad a d-b c=1$$

  Show that $G$ is a group and that it acts discontinuously on hyperbolic 3-space. Show that $G$ contains transformations that are elliptic, parabolic, hyperbolic and loxodromic.

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• Paper 4, Section II, G

  #Geometry and Groups

  2011

  Define a lattice in $\mathbb{R}^{2}$ and the rank of such a lattice.

  Let $\Lambda$ be a rank 2 lattice in $\mathbb{R}^{2}$. Choose a vector $\boldsymbol{w}_{1} \in \Lambda \backslash\{\boldsymbol{0}\}$ with $\left\|\boldsymbol{w}_{1}\right\|$ as small as possible. Then choose $\boldsymbol{w}_{2} \in \Lambda \backslash \mathbb{Z} \boldsymbol{w}_{1}$ with $\left\|\boldsymbol{w}_{2}\right\|$ as small as possible. Show that $\Lambda=\mathbb{Z} \boldsymbol{w}_{1}+\mathbb{Z} \boldsymbol{w}_{2}$.

  Suppose that $\boldsymbol{w}_{1}$ is the unit vector $\left(\begin{array}{l}1 \\ 0\end{array}\right)$. Draw the region of possible values for $\boldsymbol{w}_{2}$. Suppose that $\Lambda$ also equals $\mathbb{Z} \boldsymbol{v}_{1}+\mathbb{Z} \boldsymbol{v}_{2}$. Prove that

  $$\boldsymbol{v}_{1}=a \boldsymbol{w}_{1}+b \boldsymbol{w}_{2} \quad \text { and } \quad \boldsymbol{v}_{2}=c \boldsymbol{w}_{1}+d \boldsymbol{w}_{2}$$

  for some integers $a, b, c, d$ with $a d-b c=\pm 1$.

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• Paper 1, Section II, F

  #Graph Theory

  2011

  Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. What is a matching from $X$ to $Y$ ?

  Show that if $|\Gamma(A)| \geqslant|A|$ for all $A \subset X$ then $G$ contains a matching from $X$ to $Y$.

  Let $d$ be a positive integer. Show that if $|\Gamma(A)| \geqslant|A|-d$ for all $A \subset X$ then $G$ contains a set of $|X|-d$ independent edges.

  Show that if 0 is not an eigenvalue of $G$ then $G$ contains a matching from $X$ to $Y$.

  Suppose now that $|X|=|Y| \geqslant 1$ and that $G$ does contain a matching from $X$ to $Y$. Must it be the case that 0 is not an eigenvalue of $G$ ? Justify your answer.

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• Paper 2, Section II, F

  #Graph Theory

  2011

  What does it mean to say that a graph $G$ is $k$-colourable? Define the chromatic number $\chi(G)$ of a graph $G$, and the chromatic number $\chi(S)$ of a closed surface $S$.

  State the Euler-Poincaré formula relating the numbers of vertices, edges and faces in a drawing of a graph $G$ on a closed surface $S$ of Euler characteristic $E$. Show that if $E \leqslant 0$ then

  $$\chi(S) \leqslant\left[\frac{7+\sqrt{49-24 E}}{2}\right\rfloor$$

  Find, with justification, the chromatic number of the Klein bottle $N_{2}$. Show that if $G$ is a triangle-free graph which can be drawn on the Klein bottle then $\chi(G) \leqslant 4$.

  [You may assume that the Klein bottle has Euler characteristic 0 , and that $K_{6}$ can be drawn on the Klein bottle but $K_{7}$ cannot. You may use Brooks's theorem.]

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• Paper 3, Section II, $17 \mathrm{~F}$

  #Graph Theory

  2011

  Define the Turán graph $T_{r}(n)$. State and prove Turán's theorem. Hence, or otherwise, find $\operatorname{ex}\left(K_{3} ; n\right)$.

  Let $G$ be a bipartite graph with $n$ vertices in each class. Let $k$ be an integer, $1 \leqslant k \leqslant n$, and assume $e(G)>(k-1) n$. Show that $G$ contains a set of $k$ independent edges.

  [Hint: Suppose $G$ contains a set $D$ of a independent edges but no set of a $+1$ independent edges. Let $U$ be the set of vertices of the edges in $D$ and let $F$ be the set of edges in $G$ with precisely one vertex in $U$; consider $|F| .]$

  Hence, or otherwise, show that if $H$ is a triangle-free tripartite graph with $n$ vertices in each class then $e(H) \leqslant 2 n^{2}$.

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• Paper 4, Section II, F

  #Graph Theory

  2011

  (i) Given a positive integer $k$, show that there exists a positive integer $n$ such that, whenever the edges of the complete graph $K_{n}$ are coloured with $k$ colours, there exists a monochromatic triangle.

  Denote the least such $n$ by $f(k)$. Show that $f(k) \leqslant 3 \cdot k !$ for all $k$.

  (ii) You may now assume that $f(2)=6$ and $f(3)=17$.

  Let $H$ denote the graph of order 4 consisting of a triangle together with one extra edge. Given a positive integer $k$, let $g(k)$ denote the least positive integer $n$ such that, whenever the edges of the complete graph $K_{n}$ are coloured with $k$ colours, there exists a monochromatic copy of $H$. By considering the edges from one vertex of a monochromatic triangle in $K_{7}$, or otherwise, show that $g(2) \leqslant 7$. By exhibiting a blue-yellow colouring of the edges of $K_{6}$ with no monochromatic copy of $H$, show that in fact $g(2)=7$.

  What is $g(3) ?$ Justify your answer.

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• Paper 1, Section II, A

  #Integrable Systems

  2011

  Define a finite-dimensional integrable system and state the Arnold-Liouville theorem.

  Consider a four-dimensional phase space with coordinates $\left(q_{1}, q_{2}, p_{1}, p_{2}\right)$, where $q_{2}>0$ and $q_{1}$ is periodic with period $2 \pi$. Let the Hamiltonian be

  $$H=\frac{\left(p_{1}\right)^{2}}{2\left(q_{2}\right)^{2}}+\frac{\left(p_{2}\right)^{2}}{2}-\frac{k}{q_{2}}, \quad \text { where } k>0$$

  Show that the corresponding Hamilton equations form an integrable system.

  Determine the sign of the constant $E$ so that the motion is periodic on the surface $H=E$. Demonstrate that in this case, the action variables are given by

  $$I_{1}=p_{1}, \quad I_{2}=\gamma \int_{\alpha}^{\beta} \frac{\sqrt{\left(q_{2}-\alpha\right)\left(\beta-q_{2}\right)}}{q_{2}} d q_{2},$$

  where $\alpha, \beta, \gamma$ are positive constants which you should determine.

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• Paper 2, Section II, A

  #Integrable Systems

  2011

  Consider the Poisson structure

  $$\{F, G\}=\int_{\mathbb{R}} \frac{\delta F}{\delta u(x)} \frac{\partial}{\partial x} \frac{\delta G}{\delta u(x)} d x$$

  where $F, G$ are polynomial functionals of $u, u_{x}, u_{x x}, \ldots$. Assume that $u, u_{x}, u_{x x}, \ldots$ tend to zero as $|x| \rightarrow \infty$.

  (i) Show that $\{F, G\}=-\{G, F\}$.

  (ii) Write down Hamilton's equations for $u=u(x, t)$ corresponding to the following Hamiltonians:

  $$H_{0}[u]=\int_{\mathbb{R}} \frac{1}{2} u^{2} d x, \quad H[u]=\int_{\mathbb{R}}\left(\frac{1}{2} u_{x}^{2}+u^{3}+u u_{x}\right) d x$$

  (iii) Calculate the Poisson bracket $\left\{H_{0}, H\right\}$, and hence or otherwise deduce that the following overdetermined system of partial differential equations for $u=u\left(x, t_{0}, t\right)$ is compatible:

  $$\begin{gathered} u_{t_{0}}=u_{x} \\ u_{t}=6 u u_{x}-u_{x x x} \end{gathered}$$

  [You may assume that the Jacobi identity holds for (1).]

  (iv) Find a symmetry of (3) generated by $X=\partial / \partial u+\alpha t \partial / \partial x$ for some constant $\alpha \in \mathbb{R}$ which should be determined. Construct a vector field $Y$ corresponding to the one parameter group

  $$x \rightarrow \beta x, \quad t \rightarrow \gamma t, \quad u \rightarrow \delta u,$$

  where $(\beta, \gamma, \delta)$ should be determined from the symmetry requirement. Find the Lie algebra generated by the vector fields $(X, Y)$.

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• Paper 3, Section II, A

  #Integrable Systems

  2011

  Let $U(\rho, \tau, \lambda)$ and $V(\rho, \tau, \lambda)$ be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:

  $$\frac{\partial}{\partial \rho} \psi=U \psi, \quad \frac{\partial}{\partial \tau} \psi=V \psi$$

  where $\psi$ is a column vector whose components depend on $(\rho, \tau, \lambda)$. Using the consistency condition of this system, derive the associated zero curvature representation (ZCR)

  $$\frac{\partial}{\partial \tau} U-\frac{\partial}{\partial \rho} V+[U, V]=0,$$

  where $[\cdot, \cdot]$ denotes the usual matrix commutator.

  (i) Let

  $$U=\frac{i}{2}\left(\begin{array}{cc} 2 \lambda & \partial_{\rho} \phi \\ \partial_{\rho} \phi & -2 \lambda \end{array}\right), \quad V=\frac{1}{4 i \lambda}\left(\begin{array}{cc} \cos \phi & -i \sin \phi \\ i \sin \phi & -\cos \phi \end{array}\right)$$

  Find a partial differential equation for $\phi=\phi(\rho, \tau)$ which is equivalent to the $\mathrm{ZCR}(*)$.

  (ii) Assuming that $U$ and $V$ in $(*)$ do not depend on $t:=\rho-\tau$, show that the trace of $(U-V)^{p}$ does not depend on $x:=\rho+\tau$, where $p$ is any positive integer. Use this fact to construct a first integral of the ordinary differential equation

  $$\phi^{\prime \prime}=\sin \phi, \quad \text { where } \quad \phi=\phi(x) .$$

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• Paper 1, Section II, G

  #Linear Analysis

  2011

  State a version of the Stone-Weierstrass Theorem for real-valued functions on a compact metric space.

  Suppose that $K:[0,1]^{2} \rightarrow \mathbb{R}$ is a continuous function. Show that $K(x, y)$ may be uniformly approximated by functions of the form $\sum_{i=1}^{n} f_{i}(x) g_{i}(y)$ with $f_{i}, g_{i}:[0,1] \rightarrow \mathbb{R}$ continuous.

  Let $X, Y$ be Banach spaces and suppose that $T: X \rightarrow Y$ is a bounded linear operator. What does it mean to say that $T$ is finite-rank? What does it mean to say that $T$ is compact? Give an example of a bounded linear operator from $C[0,1]$ to itself which is not compact.

  Suppose that $\left(T_{n}\right)_{n=1}^{\infty}$ is a sequence of finite-rank operators and that $T_{n} \rightarrow T$ in the operator norm. Briefly explain why the $T_{n}$ are compact. Show that $T$ is compact.

  Hence, show that the integral operator $T: C[0,1] \rightarrow C[0,1]$ defined by

  $$T f(x)=\int_{0}^{1} f(y) K(x, y) d y$$

  is compact.

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• Paper 2, Section II, G

  #Linear Analysis

  2011

  State and prove the Baire Category Theorem. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. For $x \in \mathbb{R}$, define

  $$\omega_{f}(x)=\inf _{\delta>0} \sup _{\substack{|y-x| \leqslant \delta \\\left|y^{\prime}-x\right| \leqslant \delta}}\left|f(y)-f\left(y^{\prime}\right)\right|$$

  Show that $f$ is continuous at $x$ if and only if $\omega_{f}(x)=0$.

  Show that for any $\epsilon>0$ the set $\left\{x \in \mathbb{R}: \omega_{f}(x)<\epsilon\right\}$ is open.

  Hence show that the set of points at which $f$ is continuous cannot be precisely the set $\mathbb{Q}$ of rationals.

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• Paper 3, Section II, G

  #Linear Analysis

  2011

  Let $H$ be a complex Hilbert space with orthonormal basis $\left(e_{n}\right)_{n=-\infty}^{\infty}$ Let $T: H \rightarrow H$ be a bounded linear operator. What is meant by the spectrum $\sigma(T)$ of $T$ ?

  Define $T$ by setting $T\left(e_{n}\right)=e_{n-1}+e_{n+1}$ for $n \in \mathbb{Z}$. Show that $T$ has a unique extension to a bounded, self-adjoint linear operator on $H$. Determine the norm $\|T\|$. Exhibit, with proof, an element of $\sigma(T)$.

  Show that $T$ has no eigenvectors. Is $T$ compact?

  [General results from spectral theory may be used without proof. You may also use the fact that if a sequence $\left(x_{n}\right)$ satisfies a linear recurrence $\lambda x_{n}=x_{n-1}+x_{n+1}$ with $\lambda \in \mathbb{R}$, $|\lambda| \leqslant 2, \lambda \neq 0$, then it has the form $x_{n}=A \alpha^{n} \sin \left(\theta_{1} n+\theta_{2}\right)$ or $x_{n}=(A+n B) \alpha^{n}$, where $A, B, \alpha \in \mathbb{R}$ and $0 \leqslant \theta_{1}<\pi,\left|\theta_{2}\right| \leqslant \pi / 2$.]

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• Paper 4, Section II, G

  #Linear Analysis

  2011

  State Urysohn's Lemma. State and prove the Tietze Extension Theorem.

  Let $X, Y$ be two topological spaces. We say that the extension property holds if, whenever $S \subseteq X$ is a closed subset and $f: S \rightarrow Y$ is a continuous map, there is a continuous function $\tilde{f}: X \rightarrow Y$ with $\left.\tilde{f}\right|_{S}=f$.

  For each of the following three statements, say whether it is true or false. Briefly justify your answers.

  1. If $X$ is a metric space and $Y=[-1,1]$ then the extension property holds.

  2. If $X$ is a compact Hausdorff space and $Y=\mathbb{R}$ then the extension property holds.

  3. If $X$ is an arbitrary topological space and $Y=[-1,1]$ then the extension property holds.

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• Paper 1, Section II, H

  #Logic and Set Theory

  2011

  Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.

  Which of the following assertions about ordinals $\alpha, \beta$ and $\gamma$ are always true, and which can be false? Give proofs or counterexamples as appropriate.

  (i) $\alpha \beta=\beta \alpha$.

  (ii) $\alpha(\beta+\gamma)=\alpha \beta+\alpha \gamma$.

  (iii) If $\alpha \geqslant \omega^{2}$ then $\alpha+\omega^{2}=\omega^{2}+\alpha$.

  (iv) If $\alpha \geqslant \omega_{1}$ then $\alpha \omega_{1}=\omega_{1} \alpha$.

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• Paper 2, Section II, H

  #Logic and Set Theory

  2011

  State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your argument have you made use of the Axiom of Choice?

  Show that every real vector space has a basis.

  Let $V$ be a real vector space having a basis of cardinality $\aleph_{1}$. What is the cardinality of $V$ ? Justify your answer.

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• Paper 3, Section II, H

  #Logic and Set Theory

  2011

  State and prove the Upward Löwenheim-Skolem Theorem.

  [You may assume the Compactness Theorem, provided that you state it clearly.]

  A total ordering $(X,<)$ is called dense if for any $x<y$ there exists $z$ with $x<z<y$. Show that a dense total ordering (on more than one point) cannot be a well-ordering.

  For each of the following theories, either give axioms, in the language of posets, for the theory or prove carefully that the theory is not axiomatisable in the language of posets.

  (i) The theory of dense total orderings.

  (ii) The theory of countable dense total orderings.

  (iii) The theory of uncountable dense total orderings.

  (iv) The theory of well-orderings.

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• Paper 4, Section II, H

  #Logic and Set Theory

  2011

  Define the sets $V_{\alpha}$ for ordinals $\alpha$. Show that each $V_{\alpha}$ is transitive. Show also that $V_{\alpha} \subseteq V_{\beta}$ whenever $\alpha \leqslant \beta$. Prove that every set $x$ is a member of some $V_{\alpha}$.

  For which ordinals $\alpha$ does there exist a set $x$ such that the power-set of $x$ has rank $\alpha$ ? [You may assume standard properties of rank.]

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• Paper 1, Section I, B

  #Mathematical Biology

  2011

  A proposed model of insect dispersal is given by the equation

  $$\frac{\partial n}{\partial t}=D \frac{\partial}{\partial x}\left[\left(\frac{n_{0}}{n}\right) \frac{\partial n}{\partial x}\right]$$

  where $n(x, t)$ is the density of insects and $D$ and $n_{0}$ are constants.

  Interpret the term on the right-hand side.

  Explain why a solution of the form

  $$n(x, t)=n_{0}(D t)^{-\beta} g\left(x /(D t)^{\beta}\right),$$

  where $\beta$ is a positive constant, can potentially represent the dispersal of a fixed number $n_{0}$ of insects initially localised at the origin.

  Show that the equation (1) can be satisfied by a solution of the form (2) if $\beta=1$ and find the corresponding function $g$.

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• Paper 2, Section I, B

  #Mathematical Biology

  2011

  A population with variable growth and harvesting is modelled by the equation

  $$u_{t+1}=\max \left(\frac{r u_{t}^{2}}{1+u_{t}^{2}}-E u_{t}, 0\right)$$

  where $r$ and $E$ are positive constants.

  Given that $r>1$, show that a non-zero steady state exists if $0<E<E_{m}(r)$, where $E_{m}(r)$ is to be determined.

  Show using a cobweb diagram that, if $E<E_{m}(r)$, a non-zero steady state may be attained only if the initial population $u_{0}$ satisfies $\alpha<u_{0}<\beta$, where $\alpha$ should be determined explicitly and $\beta$ should be specified as a root of an algebraic equation.

  With reference to the cobweb diagram, give an additional criterion that implies that $\alpha<u_{0}<\beta$ is a sufficient condition, as well as a necessary condition, for convergence to a non-zero steady state.

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• Paper 3, Section I, B

  #Mathematical Biology

  2011

  The dynamics of a directly transmitted microparasite can be modelled by the system

  $$\begin{aligned} \frac{d X}{d t} &=b N-\beta X Y-b X \\ \frac{d Y}{d t} &=\beta X Y-(b+r) Y \\ \frac{d Z}{d t} &=r Y-b Z \end{aligned}$$

  where $b, \beta$ and $r$ are positive constants and $X, Y$ and $Z$ are respectively the numbers of susceptible, infected and immune (i.e. infected by the parasite, but showing no further symptoms of infection) individuals in a population of size $N$, independent of $t$, where $N=X+Y+Z$.

  Consider the possible steady states of these equations. Show that there is a threshold population size $N_{c}$ such that if $N<N_{c}$ there is no steady state with the parasite maintained in the population. Show that in this case the number of infected and immune individuals decreases to zero for all possible initial conditions.

  Show that for $N>N_{c}$ there is a possible steady state with $X=X_{s}<N$ and $Y=Y_{s}>0$, and find expressions for $X_{s}$ and $Y_{s}$.

  By linearising the equations for $d X / d t$ and $d Y / d t$ about the steady state $X=X_{s}$ and $Y=Y_{s}$, derive a quadratic equation for the possible growth or decay rate in terms of $X_{s}$ and $Y_{s}$ and hence show that the steady state is stable.

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• Paper 4, Section I, B

  #Mathematical Biology

  2011

  A neglected flower garden contains $M_{n}$ marigolds in the summer of year $n$. On average each marigold produces $\gamma$ seeds through the summer. Seeds may germinate after one or two winters. After three winters or more they will not germinate. Each winter a fraction $1-\alpha$ of all seeds in the garden are eaten by birds (with no preference to the age of the seed). In spring a fraction $\mu$ of seeds that have survived one winter and a fraction $\nu$ of seeds that have survived two winters germinate. Finite resources of water mean that the number of marigolds growing to maturity from $S$ germinating seeds is $\mathcal{N}(S)$, where $\mathcal{N}(S)$ is an increasing function such that $\mathcal{N}(0)=0, \mathcal{N}^{\prime}(0)=1, \mathcal{N}^{\prime}(S)$ is a decreasing function of $S$ and $\mathcal{N}(S) \rightarrow N_{\max }$ as $S \rightarrow \infty$

  Show that $M_{n}$ satisfies the equation

  $$M_{n+1}=\mathcal{N}\left(\alpha \mu \gamma M_{n}+\nu \gamma \alpha^{2}(1-\mu) M_{n-1}\right)$$

  Write down an equation for the number $M_{*}$ of marigolds in a steady state. Show graphically that there are two solutions, one with $M_{*}=0$ and the other with $M_{*}>0$ if

  $$\alpha \mu \gamma+\nu \gamma \alpha^{2}(1-\mu)>1$$

  Show that the $M_{*}=0$ steady-state solution is unstable to small perturbations in this case.

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• Paper 2, Section II, B

  #Mathematical Biology

  2011

  Consider a population subject to the following birth-death process. When the number of individuals in the population is $n$, the probability of an increase from $n$ to $n+1$ in unit time is $\beta n+\gamma$ and the probability of a decrease from $n$ to $n-1$ is $\alpha n(n-1)$, where $\alpha, \beta$ and $\gamma$ are constants.

  Show that the master equation for $P(n, t)$, the probability that at time $t$ the population has $n$ members, is

  $$\frac{\partial P}{\partial t}=\alpha n(n+1) P(n+1, t)-\alpha n(n-1) P(n, t)+(\beta n-\beta+\gamma) P(n-1, t)-(\beta n+\gamma) P(n, t) .$$

  Show that $\langle n\rangle$, the mean number of individuals in the population, satisfies

  $$\frac{d\langle n\rangle}{d t}=-\alpha\left\langle n^{2}\right\rangle+(\alpha+\beta)\langle n\rangle+\gamma$$

  Deduce that, in a steady state,

  $$\langle n\rangle=\frac{\alpha+\beta}{2 \alpha} \pm \sqrt{\frac{(\alpha+\beta)^{2}}{4 \alpha^{2}}+\frac{\gamma}{\alpha}-(\Delta n)^{2}}$$

  where $\Delta n$ is the standard deviation of $n$. When is the minus sign admissable?

  Show how a Fokker-Planck equation of the form

  $$\frac{\partial P}{\partial t}=\frac{\partial}{\partial n}[g(n) P(n, t)]+\frac{1}{2} \frac{\partial^{2}}{\partial n^{2}}[h(n) P(n, t)]$$

  may be derived under conditions to be explained, where the functions $g(n)$ and $h(n)$ should be evaluated.

  In the case $\alpha \ll \gamma$ and $\beta=0$, find the leading-order approximation to $n_{*}$ such that $g\left(n_{*}\right)=0$. Defining the new variable $x=n-n_{*}$, where $g\left(n_{*}\right)=0$, approximate $g(n)$ by $g^{\prime}\left(n_{*}\right) x$ and $h(n)$ by $h\left(n_{*}\right)$. Solve $(*)$ for $P(x)$ in the steady-state limit and deduce leading-order estimates for $\langle n\rangle$ and $(\Delta n)^{2}$.

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• Paper 3, Section II, B

  #Mathematical Biology

  2011

  The number density of a population of amoebae is $n(\mathbf{x}, t)$. The amoebae exhibit chemotaxis and are attracted to high concentrations of a chemical which has concentration $a(\mathbf{x}, t)$. The equations governing $n$ and $a$ are

  $$\begin{aligned} &\frac{\partial n}{\partial t}=\alpha n\left(n_{0}^{2}-n^{2}\right)+\nabla^{2} n-\nabla \cdot(\chi(n) n \nabla a) \\ &\frac{\partial a}{\partial t}=\beta n-\gamma a+D \nabla^{2} a \end{aligned}$$

  where the constants $n_{0}, \alpha, \beta, \gamma$ and $D$ are all positive.

  (i) Give a biological interpretation of each term in these equations and discuss the sign of $\chi(n)$.

  (ii) Show that there is a non-trivial (i.e. $a \neq 0, n \neq 0$ ) steady-state solution for $n$ and $a$, independent of $\mathbf{x}$, and show further that it is stable to small disturbances that are also independent of $\mathbf{x}$.

  (iii) Consider small spatially varying disturbances to the steady state, with spatial structure such that $\nabla^{2} \psi=-k^{2} \psi$, where $\psi$ is any disturbance quantity. Show that if such disturbances also satisfy $\partial \psi / \partial t=p \psi$, where $p$ is a constant, then $p$ satisfies a quadratic equation, to be derived. By considering the conditions required for $p=0$ to be a possible solution of this quadratic equation, or otherwise, deduce that instability is possible if

  $$\beta \chi_{0} n_{0}>2 \alpha n_{0}^{2} D+\gamma+2\left(2 D \alpha n_{0}^{2} \gamma\right)^{1 / 2}$$

  where $\chi_{0}=\chi\left(n_{0}\right)$.

  (iv) Explain briefly how your conclusions might change if an additional geometric constraint implied that $k^{2}>k_{0}^{2}$, where $k_{0}$ is a given constant.

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• Paper 1, Section II, F

  #Number Fields

  2011

  Calculate the class group for the field $K=\mathbb{Q}(\sqrt{-17})$.

  [You may use any general theorem, provided that you state it accurately.]

  Find all solutions in $\mathbb{Z}$ of the equation $y^{2}=x^{5}-17$.

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• Paper 2, Section II, F

  #Number Fields

  2011

  (i) Suppose that $d>1$ is a square-free integer. Describe, with justification, the ring of integers in the field $K=\mathbb{Q}(\sqrt{d})$.

  (ii) Show that $\mathbb{Q}\left(2^{1 / 3}\right)=\mathbb{Q}\left(4^{1 / 3}\right)$ and that $\mathbb{Z}\left[4^{1 / 3}\right]$ is not the ring of integers in this field.

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• Paper 4, Section II, F

  #Number Fields

  2011

  (i) Prove that the ring of integers $\mathcal{O}_{K}$ in a real quadratic field $K$ contains a non-trivial unit. Any general results about lattices and convex bodies may be assumed.

  (ii) State the general version of Dirichlet's unit theorem.

  (iii) Show that for $K=\mathbb{Q}(\sqrt{7}), 8+3 \sqrt{7}$ is a fundamental unit in $\mathcal{O}_{K}$.

  [You may not use results about continued fractions unless you prove them.]

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• Paper 1, Section I, I

  #Number Theory

  2011

  Prove that, under the action of $\mathrm{SL}_{2}(\mathbb{Z})$, every positive definite binary quadratic form of discriminant $-163$, with integer coefficients, is equivalent to

  $$x^{2}+x y+41 y^{2} .$$

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• Paper 4, Section $I$, I

  #Number Theory

  2011

  (i) Prove that there are infinitely many primes.

  (ii) Prove that arbitrarily large gaps can occur between consecutive primes.

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• Paper 2, Section I, I

  #Number Theory

  2011

  (i) Find a primitive root modulo $17 .$

  (ii) Let $p$ be a prime of the form $2^{m}+1$ for some integer $m \geqslant 1$. Prove that every quadratic non-residue modulo $p$ is a primitive root modulo $p$.

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• Paper 3, Section I, I

  #Number Theory

  2011

  (i) State Lagrange's Theorem, and prove that, if $p$ is an odd prime,

  $$(p-1) ! \equiv-1 \quad \bmod p$$

  (ii) Still assuming $p$ is an odd prime, prove that

  $$3^{2} \cdot 5^{2} \cdots(p-2)^{2} \equiv(-1)^{\frac{p+1}{2}} \bmod p$$

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• Paper 3, Section II, I

  #Number Theory

  2011

  Let $\zeta(s)$ be the Riemann zeta function, and put $s=\sigma+i t$ with $\sigma, t \in \mathbb{R}$.

  (i) If $\sigma>1$, prove that

  $$\zeta(s)=\prod_{p}\left(1-p^{-s}\right)^{-1}$$

  where the product is taken over all primes $p$.

  (ii) Assuming that, for $\sigma>1$, we have

  $$\zeta(s)=\sum_{n=1}^{\infty} n\left(n^{-s}-(n+1)^{-s}\right)$$

  prove that $\zeta(s)-\frac{1}{s-1}$ has an analytic continuation to the half plane $\sigma>0$.

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• Paper 4, Section II, I

  #Number Theory

  2011

  (i) Prove the law of reciprocity for the Jacobi symbol. You may assume the law of reciprocity for the Legendre symbol.

  (ii) Let $n$ be an odd positive integer which is not a square. Prove that there exists an odd prime $p$ with $\left(\frac{n}{p}\right)=-1$.

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• Paper 1, Section II, A

  #Numerical Analysis

  2011

  The nine-point method for the Poisson equation $\nabla^{2} u=f$ (with zero Dirichlet boundary conditions) in a square, reads

  $$\begin{array}{r} \frac{2}{3}\left(u_{i-1, j}+u_{i+1, j}+u_{i, j-1}+u_{i, j+1}\right)+\frac{1}{6}\left(u_{i-1, j-1}+u_{i-1, j+1}+u_{i+1, j-1}+u_{i+1, j+1}\right) \\ -\frac{10}{3} u_{i, j}=h^{2} f_{i, j}, \quad i, j=1, \ldots, m, \end{array}$$

  where $u_{0, j}=u_{m+1, j}=u_{i, 0}=u_{i, m+1}=0$, for all $i, j=0, \ldots, m+1$.

  (i) By arranging the two-dimensional arrays $\left\{u_{i, j}\right\}_{i, j=1, \ldots, m}$ and $\left\{f_{i, j}\right\}_{i, j=1, \ldots, m}$ into column vectors $u \in \mathbb{R}^{m^{2}}$ and $b \in \mathbb{R}^{m^{2}}$ respectively, the linear system above takes the matrix form $A u=b$. Prove that, regardless of the ordering of the points on the grid, the matrix $A$ is symmetric and negative definite.

  (ii) Formulate the Jacobi method with relaxation for solving the above linear system.

  (iii) Prove that the iteration converges if the relaxation parameter $\omega$ is equal to $1 .$

  [You may quote without proof any relevant result about convergence of iterative methods.]

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• Paper 2, Section II, A

  #Numerical Analysis

  2011

  Let $A \in \mathbb{R}^{n \times n}$ be a real matrix with $n$ linearly independent eigenvectors. The eigenvalues of $A$ can be calculated from the sequence $x^{(k)}, k=0,1, \ldots$, which is generated by the power method

  $$x^{(k+1)}=\frac{A x^{(k)}}{\left\|A x^{(k)}\right\|},$$

  where $x^{(0)}$ is a real nonzero vector.

  (i) Describe the asymptotic properties of the sequence $x^{(k)}$ in the case that the eigenvalues $\lambda_{i}$ of $A$ satisfy $\left|\lambda_{i}\right|<\left|\lambda_{n}\right|, i=1, \ldots, n-1$, and the eigenvectors are of unit length.

  (ii) Present the implementation details for the power method for the setting in (i) and define the Rayleigh quotient.

  (iii) Let $A$ be the $3 \times 3$ matrix

  $$A=\lambda I+P, \quad P=\left(\begin{array}{lll} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right)$$

  where $\lambda$ is real and nonzero. Find an explicit expression for $A^{k}, k=1,2,3, \ldots$

  Let the sequence $x^{(k)}$ be generated by the power method as above. Deduce from your expression for $A^{k}$ that the first and second components of $x^{(k+1)}$ tend to zero as $k \rightarrow \infty$. Further show that this implies $A x^{(k+1)}-\lambda x^{(k+1)} \rightarrow 0$ as $k \rightarrow \infty$.

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• Paper 3, Section II, A

  #Numerical Analysis

  2011

  (i) The difference equation

  $$u_{i}^{n+1}=u_{i}^{n}+\frac{3}{2} \mu\left(u_{i-1}^{n}-2 u_{i}^{n}+u_{i+1}^{n}\right)-\frac{1}{2} \mu\left(u_{i-1}^{n-1}-2 u_{i}^{n-1}+u_{i+1}^{n-1}\right)$$

  where $\mu=\Delta t /(\Delta x)^{2}$, is the basic equation used in the second-order AdamsBashforth method and can be employed to approximate a solution of the diffusion equation $u_{t}=u_{x x}$. Prove that, as $\Delta t \rightarrow 0$ with constant $\mu$, the local error of the method is $O(\Delta t)^{2}$.

  (ii) By applying the Fourier stability test, show that the above method is stable if and only if $\mu \leqslant 1 / 4$.

  (iii) Define the leapfrog scheme to approximate the diffusion equation and prove that it is unstable for every choice of $\mu>0$.

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• Paper 4, Section II, A

  #Numerical Analysis

  2011

  (i) Consider the Poisson equation

  $$\nabla^{2} u=f, \quad-1 \leqslant x, y \leqslant 1$$

  with the periodic boundary conditions

  and the normalization condition

  $$\int_{-1}^{1} \int_{-1}^{1} u(x, y) d x d y=0$$

  Moreover, $f$ is analytic and obeys the periodic boundary conditions $f(-1, y)=$ $f(1, y), f(x,-1)=f(x, 1),-1 \leqslant x, y \leqslant 1 .$

  Derive an explicit expression of the approximation of a solution $u$ by means of a spectral method. Explain the term convergence with spectral speed and state its validity for the approximation of $u$.

  (ii) Consider the second-order linear elliptic partial differential equation

  $$\nabla \cdot(a \nabla u)=f, \quad-1 \leqslant x, y \leqslant 1$$

  with the periodic boundary conditions and normalization condition specified in (i). Moreover, $a$ and $f$ are given by

  $$a(x, y)=\cos (\pi x)+\cos (\pi y)+3, \quad f(x, y)=\sin (\pi x)+\sin (\pi y)$$

  [Note that $a$ is a positive analytic periodic function.]

  Construct explicitly the linear algebraic system that arises from the implementation of a spectral method to the above equation.

  $$\begin{aligned} & u(-1, y)=u(1, y), \quad u_{x}(-1, y)=u_{x}(1, y), \quad-1 \leqslant y \leqslant 1, \\ & u(x,-1)=u(x, 1), \quad u_{y}(x,-1)=u_{y}(x, 1), \quad-1 \leqslant x \leqslant 1 \end{aligned}$$

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• Paper 2, Section II, K

  #Optimization and Control

  2011

  Consider an optimal stopping problem in which the optimality equation takes the form

  $$F_{t}(x)=\max \left\{r(x), E\left[F_{t+1}\left(x_{t+1}\right)\right]\right\}, \quad t=1, \ldots, N-1,$$

  $F_{N}(x)=r(x)$, and where $r(x)>0$ for all $x$. Let $S$ denote the stopping set of the onestep-look-ahead rule. Show that if $S$ is closed (in a sense you should explain) then the one-step-look-ahead rule is optimal.

  $N$ biased coins are to be tossed successively. The probability that the $i$ th coin toss will show a head is known to be $p_{i}\left(0<p_{i}<1\right)$. At most once, after observing a head, and before tossing the next coin, you may guess that you have just seen the last head (i.e. that all subsequent tosses will show tails). If your guess turns out to be correct then you win $£ 1$.

  Suppose that you have not yet guessed 'last head', and the $i$ th toss is a head. Show that it cannot be optimal to guess that this is the last head if

  $$\frac{p_{i+1}}{q_{i+1}}+\cdots+\frac{p_{N}}{q_{N}}>1$$

  where $q_{j}=1-p_{j}$.

  Suppose that $p_{i}=1 / i$. Show that it is optimal to guess that the last head is the first head (if any) to occur after having tossed at least $i^{*}$ coins, where $i^{*} \approx N / e$ when $N$ is large.

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• Paper 3, Section II, 28K

  #Optimization and Control

  2011

  An observable scalar state variable evolves as $x_{t+1}=x_{t}+u_{t}, t=0,1, \ldots$ Let controls $u_{0}, u_{1}, \ldots$ be determined by a policy $\pi$ and define

  $$C_{s}\left(\pi, x_{0}\right)=\sum_{t=0}^{s-1}\left(x_{t}^{2}+2 x_{t} u_{t}+7 u_{t}^{2}\right) \quad \text { and } \quad C_{s}\left(x_{0}\right)=\inf _{\pi} C_{s}\left(\pi, x_{0}\right)$$

  Show that it is possible to express $C_{s}\left(x_{0}\right)$ in terms of $\Pi_{s}$, which satisfies the recurrence

  $$\Pi_{s}=\frac{6\left(1+\Pi_{s-1}\right)}{7+\Pi_{s-1}}, \quad s=1,2, \ldots$$

  with $\Pi_{0}=0$.

  Deduce that $C_{\infty}\left(x_{0}\right) \geqslant 2 x_{0}^{2} .\left[C_{\infty}\left(x_{0}\right)\right.$ is defined as $\left.\lim _{s \rightarrow \infty} C_{s}\left(x_{0}\right) .\right]$

  By considering the policy $\pi^{*}$ which takes $u_{t}=-(1 / 3)(2 / 3)^{t} x_{0}, t=0,1, \ldots$, show that $C_{\infty}\left(x_{0}\right)=2 x_{0}^{2}$.

  Give an alternative description of $\pi^{*}$ in closed-loop form.

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• Paper 4, Section II, K

  #Optimization and Control

  2011

  Describe the type of optimal control problem that is amenable to analysis using Pontryagin's Maximum Principle.

  A firm has the right to extract oil from a well over the interval $[0, T]$. The oil can be sold at price $£ p$ per unit. To extract oil at rate $u$ when the remaining quantity of oil in the well is $x$ incurs cost at rate $£ u^{2} / x$. Thus the problem is one of maximizing

  $$\int_{0}^{T}\left[p u(t)-\frac{u(t)^{2}}{x(t)}\right] d t$$

  subject to $d x(t) / d t=-u(t), u(t) \geqslant 0, x(t) \geqslant 0$. Formulate the Hamiltonian for this problem.

  Explain why $\lambda(t)$, the adjoint variable, has a boundary condition $\lambda(T)=0$.

  Use Pontryagin's Maximum Principle to show that under optimal control

  $$\lambda(t)=p-\frac{1}{1 / p+(T-t) / 4}$$

  and

  $$\frac{d x(t)}{d t}=-\frac{2 p x(t)}{4+p(T-t)}$$

  Find the oil remaining in the well at time $T$, as a function of $x(0), p$, and $T$,

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• Paper 1, Section II, A

  #Partial Differential Equations

  2011

  Let $H=H(x, v), x, v \in \mathbb{R}^{n}$, be a smooth real-valued function which maps $\mathbb{R}^{2 n}$ into $\mathbb{R}$. Consider the initial value problem for the equation

  $$\begin{aligned} &f_{t}+\nabla_{v} H \cdot \nabla_{x} f-\nabla_{x} H \cdot \nabla_{v} f=0, \quad x, v \in \mathbb{R}^{n}, t>0 \\ &f(x, v, t=0)=f_{I}(x, v), \quad x, v \in \mathbb{R}^{n} \end{aligned}$$

  for the unknown function $f=f(x, v, t)$.

  (i) Use the method of characteristics to solve the initial value problem, locally in time.

  (ii) Let $f_{I} \geqslant 0$ on $\mathbb{R}^{2 n}$. Use the method of characteristics to prove that $f$ remains non-negative (as long as it exists).

  (iii) Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be smooth. Prove that

  $$\int_{\mathbb{R}^{2 n}} F(f(x, v, t)) d x d v=\int_{\mathbb{R}^{2 n}} F\left(f_{I}(x, v)\right) d x d v$$

  as long as the solution exists.

  (iv) Let $H$ be independent of $x$, namely $H(x, v)=a(v)$, where $a$ is smooth and realvalued. Give the explicit solution of the initial value problem.

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• Paper 2, Section II, A

  #Partial Differential Equations

  2011

  Consider the Schrödinger equation

  $$\begin{aligned} i \partial_{t} \psi(t, x) &=-\frac{1}{2} \Delta \psi(t, x)+V(x) \psi(t, x), \quad x \in \mathbb{R}^{n}, t>0 \\ \psi(t=0, x) &=\psi_{I}(x), \quad x \in \mathbb{R}^{n} \end{aligned}$$

  where $V$ is a smooth real-valued function.

  Prove that, for smooth solutions, the following equations are valid for all $t>0$ :

  (i)

  $$\int_{\mathbb{R}^{n}}|\psi(t, x)|^{2} d x=\int_{\mathbb{R}^{n}}\left|\psi_{I}(x)\right|^{2} d x$$

  (ii)

  $$\begin{aligned} &\int_{\mathbb{R}^{n}} \frac{1}{2}|\nabla \psi(t, x)|^{2} d x+\int_{\mathbb{R}^{n}} V(x)|\psi(t, x)|^{2} d x \\ &=\int_{\mathbb{R}^{n}} \frac{1}{2}\left|\nabla \psi_{I}(x)\right|^{2} d x+\int_{\mathbb{R}^{n}} V(x)\left|\psi_{I}(x)\right|^{2} d x \end{aligned}$$

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• Paper 3, Section II, A

  #Partial Differential Equations

  2011

  (a) State the local existence theorem of a classical solution of the Cauchy problem

  $$\begin{aligned} &a\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{1}}+b\left(x_{1}, x_{2}, u\right) \frac{\partial u}{\partial x_{2}}=c\left(x_{1}, x_{2}, u\right) \\ &\left.u\right|_{\Gamma}=u_{0} \end{aligned}$$

  where $\Gamma$ is a smooth curve in $\mathbb{R}^{2}$.

  (b) Solve, by using the method of characteristics,

  $$\begin{aligned} &2 x_{1} \frac{\partial u}{\partial x_{1}}+4 x_{2} \frac{\partial u}{\partial x_{2}}=u^{2} \\ &u\left(x_{1}, 2\right)=h \end{aligned}$$

  where $h>0$ is a constant. What is the maximal domain of existence in which $u$ is a solution of the Cauchy problem?

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• Paper 4, Section II, A

  #Partial Differential Equations

  2011

  Consider the functional

  $$E(u)=\frac{1}{2} \int_{\Omega}|\nabla u|^{2} d x+\int_{\Omega} F(u, x) d x$$

  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $F: \mathbb{R} \times \Omega \rightarrow \mathbb{R}$ is smooth. Assume that $F(u, x)$ is convex in $u$ for all $x \in \Omega$ and that there is a $K>0$ such that

  $$-K \leqslant F(v, x) \leqslant K\left(|v|^{2}+1\right) \quad \forall v \in \mathbb{R}, x \in \Omega$$

  (i) Prove that $E$ is well-defined on $H_{0}^{1}(\Omega)$, bounded from below and strictly convex. Assume without proof that $E$ is weakly lower-semicontinuous. State this property. Conclude the existence of a unique minimizer of $E$.

  (ii) Which elliptic boundary value problem does the minimizer solve?

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• Paper 1, Section II, D

  #Principles of Quantum Mechanics

  2011

  Two individual angular momentum states $\left|j_{1}, m_{1}\right\rangle,\left|j_{2}, m_{2}\right\rangle$, acted on by $\mathbf{J}^{(1)}$ and $\mathbf{J}^{(2)}$ respectively, can be combined to form a combined state $|J, M\rangle$. What is the combined angular momentum operator $\mathbf{J}$ in terms of $\mathbf{J}^{(1)}$ and $\mathbf{J}^{(2)}$ ? [Units in which $\hbar=1$ are to be used throughout.]

  Defining raising and lowering operators $J_{\pm}^{(i)}$, where $i \in\{1,2\}$, find an expression for $\mathbf{J}^{2}$ in terms of $\mathbf{J}^{(i)^{2}}, J_{\pm}^{(i)}$ and $J_{3}^{(i)}$. Show that this implies

  $$\left[\mathbf{J}^{2}, J_{3}\right]=0$$

  Write down the state with $J=j_{1}+j_{2}$ and with $J_{3}$ eigenvalue $M=-j_{1}-j_{2}$ in terms of the individual angular momentum states. From this starting point, calculate the combined state with eigenvalues $J=j_{1}+j_{2}-1$ and $M=-j_{1}-j_{2}+1$ in terms of the individual angular momentum states.

  If $j_{1}=3$ and $j_{2}=1$ and the combined system is in the state $|3,-3\rangle$, what is the probability of measuring the $J_{3}^{(i)}$ eigenvalues of individual angular momentum states to be $-3$ and 0 , respectively?

  [You may assume without proof that standard angular momentum states $|j, m\rangle$ are joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, obeying

  $$J_{\pm}|j, m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j, m+1\rangle,$$

  and that

  $$\left.\left[J_{\pm}, J_{3}\right]=\pm J_{\pm} \cdot\right]$$

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• Paper 2, Section II, D

  #Principles of Quantum Mechanics

  2011

  A quantum system has energy eigenstates $|n\rangle$ with eigenvalues $E_{n}=n \hbar, n \in$ $\{1,2,3, \ldots\}$. An observable $Q$ is such that $Q|n\rangle=q_{n}|n\rangle$.

  (a) What is the commutator of $Q$ with the Hamiltonian $H$ ?

  (b) Given $q_{n}=\frac{1}{n}$, consider the state

  $$|\psi\rangle \propto \sum_{n=1}^{N} \sqrt{n}|n\rangle$$

  Determine:

  (i) The probability of measuring $Q$ to be $1 / N$.

  (ii) The probability of measuring energy $\hbar$ followed by another immediate measurement of energy $2 \hbar$.

  (iii) The average of many separate measurements of $Q$, each measurement being on a state $|\psi\rangle$, as $N \rightarrow \infty$.

  (c) Given $q_{1}=1$ and $q_{n}=-1$ for $n>1$, consider the state

  $$|\psi\rangle \propto \sum_{n=1}^{\infty} \alpha^{n / 2}|n\rangle,$$

  where $0<\alpha<1$.

  (i) Show that the probability of measuring an eigenvalue $q=-1$ of $|\psi\rangle$ is

  $$A+B \alpha$$

  where $A$ and $B$ are integers that you should find.

  (ii) Show that $\langle Q\rangle_{\psi}$ is $C+D \alpha$, where $C$ and $D$ are integers that you should find.

  (iii) Given that $Q$ is measured to be $-1$ at time $t=0$, write down the state after a time $t$ has passed. What is then the subsequent probability at time $t$ of measuring the energy to be $2 \hbar$ ?

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• Paper 3, Section II, D

  #Principles of Quantum Mechanics

  2011

  The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right)=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$, with

  $$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

  are used to represent angular momentum operators with respect to basis states $|\uparrow\rangle$ and $|\downarrow\rangle$ corresponding to spin up and spin down along the $z$-axis. They satisfy

  $$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$$

  (i) How are $|\uparrow\rangle$ and $|\downarrow\rangle$ represented? How is the spin operator s related to $\sigma$ and $\hbar$ ? Check that the commutation relations between the spin operators are as desired. Check that $\mathbf{s}^{2}$ acting on a spin one-half state has the correct eigenvalue.

  What are the states obtained by applying $s_{x}, s_{y}$ to the eigenstates $|\uparrow\rangle$ and $|\downarrow\rangle$ of $s_{z}$ ?

  (ii) Let $V$ be the space of states for a spin one-half system. Consider a combination of three such systems with states belonging to $V^{(1)} \otimes V^{(2)} \otimes V^{(3)}$ and spin operators acting on each subsystem denoted by $s_{x}^{(i)}, s_{y}^{(i)}$ with $i=1,2,3$. Find the eigenvalues of the operators

  $$s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)} \quad \text { and } \quad s_{x}^{(1)} s_{x}^{(2)} s_{x}^{(3)}$$

  of the state

  $$|\Psi\rangle=\frac{1}{\sqrt{2}}\left[|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}-|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right]$$

  (iii) Consider now whether these outcomes for measurements of particular combinations of the operators $s_{x}^{(i)}, s_{y}^{(i)}$ in the state $|\Psi\rangle$ could be reproduced by replacing the spin operators with classical variables $\tilde{s}_{x}^{(i)}, \tilde{s}_{y}^{(i)}$ which take values $\pm \hbar / 2$ according to some probabilities. Assume that these variables are identical to the quantum measurements of $s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)}$ on $|\Psi\rangle$. Show that classically this implies a unique possibility for

  $$\tilde{s}_{x}^{(1)} \tilde{s}_{x}^{(2)} \tilde{s}_{x}^{(3)},$$

  and find its value.

  State briefly how this result could be used to experimentally test quantum mechanics.

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• Paper 4, Section II, D

  #Principles of Quantum Mechanics

  2011

  The quantum-mechanical observable $Q$ has just two orthonormal eigenstates $|1\rangle$ and $|2\rangle$ with eigenvalues $-1$ and 1 , respectively. The operator $Q^{\prime}$ is defined by $Q^{\prime}=Q+\epsilon T$, where

  $$T=\left(\begin{array}{cc} 0 & i \\ -i & 0 \end{array}\right)$$

  Defining orthonormal eigenstates of $Q^{\prime}$ to be $\left|1^{\prime}\right\rangle$ and $\left|2^{\prime}\right\rangle$ with eigenvalues $q_{1}^{\prime}$, $q_{2}^{\prime}$, respectively, consider a perturbation to first order in $\epsilon \in \mathbb{R}$ for the states

  $$\left|1^{\prime}\right\rangle=a_{1}|1\rangle+a_{2} \epsilon|2\rangle, \quad\left|2^{\prime}\right\rangle=b_{1}|2\rangle+b_{2} \epsilon|1\rangle,$$

  where $a_{1}, a_{2}, b_{1}, b_{2}$ are complex coefficients. The real eigenvalues are also expanded to first order in $\epsilon$ :

  $$q_{1}^{\prime}=-1+c_{1} \epsilon, \quad q_{2}^{\prime}=1+c_{2} \epsilon$$

  From first principles, find $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}$.

  Working exactly to all orders, find the real eigenvalues $q_{1}^{\prime}, q_{2}^{\prime}$ directly. Show that the exact eigenvectors of $Q^{\prime}$ may be taken to be of the form

  $$A_{j}(\epsilon)\left(\begin{array}{c} 1 \\ -i\left(1+B q_{j}^{\prime}\right) / \epsilon \end{array}\right)$$

  finding $A_{j}(\epsilon)$ and the real numerical coefficient $B$ in the process.

  By expanding the exact expressions, again find $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}$, verifying the perturbation theory results above.

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• Paper 1, Section II, K

  #Principles of Statistics

  2011

  Define admissible, Bayes, minimax decision rules.

  A random vector $X=\left(X_{1}, X_{2}, X_{3}\right)^{\mathrm{T}}$ has independent components, where $X_{i}$ has the normal distribution $\mathcal{N}\left(\theta_{i}, 1\right)$ when the parameter vector $\Theta$ takes the value $\theta=\left(\theta_{1}, \theta_{2}, \theta_{3}\right)^{\mathrm{T}}$. It is required to estimate $\Theta$ by a point $a \in \mathbb{R}^{3}$, with loss function $L(\theta, a)=\|a-\theta\|^{2}$. What is the risk function of the maximum-likelihood estimator $\widehat{\Theta}:=X ?$ Show that $\widehat{\Theta}$ is dominated by the estimator $\widetilde{\Theta}:=\left(1-\|X\|^{-2}\right) X$.

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• Paper 2, Section II, K

  #Principles of Statistics

  2011

  Random variables $X_{1}, \ldots, X_{n}$ are independent and identically distributed from the normal distribution with unknown mean $\mathrm{M}$ and unknown precision (inverse variance) $H$. Show that the likelihood function, for data $X_{1}=x_{1}, \ldots, X_{n}=x_{n}$, is

  $$L_{n}(\mu, h) \propto h^{n / 2} \exp \left(-\frac{1}{2} h\left\{n(\bar{x}-\mu)^{2}+S\right\}\right)$$

  where $\bar{x}:=n^{-1} \sum_{i} x_{i}$ and $S:=\sum_{i}\left(x_{i}-\bar{x}\right)^{2}$.

  A bivariate prior distribution for $(\mathrm{M}, H)$ is specified, in terms of hyperparameters $\left(\alpha_{0}, \beta_{0}, m_{0}, \lambda_{0}\right)$, as follows. The marginal distribution of $H$ is $\Gamma\left(\alpha_{0}, \beta_{0}\right)$, with density

  $$\pi(h) \propto h^{\alpha_{0}-1} e^{-\beta_{0} h} \quad(h>0),$$

  and the conditional distribution of $\mathrm{M}$, given $H=h$, is normal with mean $m_{0}$ and precision $\lambda_{0} h$.

  Show that the conditional prior distribution of $H$, given $M=\mu$, is

  $$H \mid \mathrm{M}=\mu \quad \sim\left(\alpha_{0}+\frac{1}{2}, \beta_{0}+\frac{1}{2} \lambda_{0}\left(\mu-m_{0}\right)^{2}\right)$$

  Show that the posterior joint distribution of $(\mathrm{M}, H)$, given $X_{1}=x_{1}, \ldots, X_{n}=x_{n}$, has the same form as the prior, with updated hyperparameters $\left(\alpha_{n}, \beta_{n}, m_{n}, \lambda_{n}\right)$ which you should express in terms of the prior hyperparameters and the data.

  [You may use the identity

  $$p(t-a)^{2}+q(t-b)^{2}=(t-\delta)^{2}+p q(a-b)^{2}$$

  where $p+q=1$ and $\delta=p a+q b$.]

  Explain how you could implement Gibbs sampling to generate a random sample from the posterior joint distribution.

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• Paper 3, Section II, $27 \mathrm{~K}$

  #Principles of Statistics

  2011

  Random variables $X_{1}, X_{2}, \ldots$ are independent and identically distributed from the exponential distribution $\mathcal{E}(\theta)$, with density function

  $$p_{X}(x \mid \theta)=\theta e^{-\theta x} \quad(x>0),$$

  when the parameter $\Theta$ takes value $\theta>0$. The following experiment is performed. First $X_{1}$ is observed. Thereafter, if $X_{1}=x_{1}, \ldots, X_{i}=x_{i}$ have been observed $(i \geqslant 1)$, a coin having probability $\alpha\left(x_{i}\right)$ of landing heads is tossed, where $\alpha: \mathbb{R} \rightarrow(0,1)$ is a known function and the coin toss is independent of the $X$ 's and previous tosses. If it lands heads, no further observations are made; if tails, $X_{i+1}$ is observed.

  Let $N$ be the total number of $X$ 's observed, and $\mathbf{X}:=\left(X_{1}, \ldots, X_{N}\right)$. Write down the likelihood function for $\Theta$ based on data $\mathbf{X}=\left(x_{1}, \ldots, x_{n}\right)$, and identify a minimal sufficient statistic. What does the likelihood principle have to say about inference from this experiment?

  Now consider the experiment that only records $Y:=X_{N}$. Show that the density function of $Y$ has the form

  $$p_{Y}(y \mid \theta)=\exp \{a(y)-k(\theta)-\theta y\}$$

  Assuming the function $a(\cdot)$ is twice differentiable and that both $p_{Y}(y \mid \theta)$ and $\partial p_{Y}(y \mid \theta) / \partial y$ vanish at 0 and $\infty$, show that $a^{\prime}(Y)$ is an unbiased estimator of $\Theta$, and find its variance.

  Stating clearly any general results you use, deduce that

  $$-k^{\prime \prime}(\theta) \mathbb{E}_{\theta}\left\{a^{\prime \prime}(Y)\right\} \geqslant 1 .$$

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• Paper 4, Section II, K

  #Principles of Statistics

  2011

  What does it mean to say that a $(1 \times p)$ random vector $\xi$ has a multivariate normal distribution?

  Suppose $\xi=(X, Y)$ has the bivariate normal distribution with mean vector $\mu=\left(\mu_{X}, \mu_{Y}\right)$, and dispersion matrix

  $$\Sigma=\left(\begin{array}{cc} \sigma_{X X} & \sigma_{X Y} \\ \sigma_{X Y} & \sigma_{Y Y} \end{array}\right)$$

  Show that, with $\beta:=\sigma_{X Y} / \sigma_{X X}, Y-\beta X$ is independent of $X$, and thus that the conditional distribution of $Y$ given $X$ is normal with mean $\mu_{Y}+\beta\left(X-\mu_{X}\right)$ and variance $\sigma_{Y Y \cdot X}:=\sigma_{Y Y}-\sigma_{X Y}^{2} / \sigma_{X X}$.

  For $i=1, \ldots, n, \xi_{i}=\left(X_{i}, Y_{i}\right)$ are independent and identically distributed with the above distribution, where all elements of $\mu$ and $\Sigma$ are unknown. Let

  $$S=\left(\begin{array}{cc} S_{X X} & S_{X Y} \\ S_{X Y} & S_{Y Y} \end{array}\right):=\sum_{i=1}^{n}\left(\xi_{i}-\bar{\xi}\right)^{\mathrm{T}}\left(\xi_{i}-\bar{\xi}\right)$$

  where $\bar{\xi}:=n^{-1} \sum_{i=1}^{n} \xi_{i}$.

  The sample correlation coefficient is $r:=S_{X Y} / \sqrt{S_{X X} S_{Y Y}}$. Show that the distribution of $r$ depends only on the population correlation coefficient $\rho:=\sigma_{X Y} / \sqrt{\sigma_{X X} \sigma_{Y Y}}$.

  Student's $t$-statistic (on $n-2$ degrees of freedom) for testing the null hypothesis $H_{0}: \beta=0$ is

  $$t:=\frac{\widehat{\beta}}{\sqrt{S_{Y Y \cdot X} /(n-2) S_{X X}}},$$

  where $\widehat{\beta}:=S_{X Y} / S_{X X}$ and $S_{Y Y \cdot X}:=S_{Y Y}-S_{X Y}^{2} / S_{X X}$. Its density when $H_{0}$ is true is

  $$p(t)=C\left(1+\frac{t^{2}}{n-2}\right)^{-\frac{1}{2}(n-1)}$$

  where $C$ is a constant that need not be specified.

  Express $t$ in terms of $r$, and hence derive the density of $r$ when $\rho=0$.

  How could you use the sample correlation $r$ to test the hypothesis $\rho=0$ ?

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• Paper 1, Section II, $26 \mathrm{~K}$

  #Probability and Measure

  2011

  (i) Let $(E, \mathcal{E}, \mu)$ be a measure space and let $1 \leqslant p<\infty$. For a measurable function $f$, let $\|f\|_{p}=\left(\int|f|^{p} d \mu\right)^{1 / p}$. Give the definition of the space $L^{p}$. Prove that $\left(L^{p},\|\cdot\|_{p}\right)$ forms a Banach space.

  [You may assume that $L^{p}$ is a normed vector space. You may also use in your proof any other result from the course provided that it is clearly stated.]

  (ii) Show that convergence in probability implies convergence in distribution.

  [Hint: Show the pointwise convergence of the characteristic function, using without proof the inequality $\left|e^{i y}-e^{i x}\right| \leqslant|x-y|$ for $x, y \in \mathbb{R}$.]

  (iii) Let $\left(\alpha_{j}\right)_{j \geqslant 1}$ be a given real-valued sequence such that $\sum_{j=1}^{\infty} \alpha_{j}^{2}=\sigma^{2}<\infty$. Let $\left(X_{j}\right)_{j \geqslant 1}$ be a sequence of independent standard Gaussian random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let

  $$Y_{n}=\sum_{j=1}^{n} \alpha_{j} X_{j}$$

  Prove that there exists a random variable $Y$ such that $Y_{n} \rightarrow Y$ in $L^{2}$.

  (iv) Specify the distribution of the random variable $Y$ defined in part (iii), justifying carefully your answer.

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• Paper 2, Section II, $26 \mathrm{~K}$

  #Probability and Measure

  2011

  (i) Define the notions of a $\pi$-system and a $d$-system. State and prove Dynkin's lemma.

  (ii) Let $\left(E_{1}, \mathcal{E}_{1}, \mu_{1}\right)$ and $\left(E_{2}, \mathcal{E}_{2}, \mu_{2}\right)$ denote two finite measure spaces. Define the $\sigma$ algebra $\mathcal{E}_{1} \otimes \mathcal{E}_{2}$ and the product measure $\mu_{1} \otimes \mu_{2}$. [You do not need to verify that such a measure exists.] State (without proof) Fubini's Theorem.

  (iii) Let $(E, \mathcal{E}, \mu)$ be a measure space, and let $f$ be a non-negative Borel-measurable function. Let $G$ be the subset of $E \times \mathbb{R}$ defined by

  $$G=\{(x, y) \in E \times \mathbb{R}: 0 \leqslant y \leqslant f(x)\}$$

  Show that $G \in \mathcal{E} \otimes \mathcal{B}(\mathbb{R})$, where $\mathcal{B}(\mathbb{R})$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Show further that

  $$\int f d \mu=(\mu \otimes \lambda)(G)$$

  where $\lambda$ is Lebesgue measure.

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• Paper 3, Section II, 25K

  #Probability and Measure

  2011

  (i) State and prove Kolmogorov's zero-one law.

  (ii) Let $(E, \mathcal{E}, \mu)$ be a finite measure space and suppose that $\left(B_{n}\right)_{n \geqslant 1}$ is a sequence of events such that $B_{n+1} \subset B_{n}$ for all $n \geqslant 1$. Show carefully that $\mu\left(B_{n}\right) \rightarrow \mu(B)$, where $B=\cap_{n=1}^{\infty} B_{n}$.

  (iii) Let $\left(X_{i}\right)_{i \geqslant 1}$ be a sequence of independent and identically distributed random variables such that $\mathbb{E}\left(X_{1}^{2}\right)=\sigma^{2}<\infty$ and $\mathbb{E}\left(X_{1}\right)=0$. Let $K>0$ and consider the event $A_{n}$ defined by

  $$A_{n}=\left\{\frac{S_{n}}{\sqrt{n}} \geqslant K\right\}, \quad \text { where } \quad S_{n}=\sum_{i=1}^{n} X_{i}$$

  Prove that there exists $c>0$ such that for all $n$ large enough, $\mathbb{P}\left(A_{n}\right) \geqslant c$. Any result used in the proof must be stated clearly.

  (iv) Prove using the results above that $A_{n}$ occurs infinitely often, almost surely. Deduce that

  $$\limsup _{n \rightarrow \infty} \frac{S_{n}}{\sqrt{n}}=\infty$$

  almost surely.

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• Paper 4, Section II, K

  #Probability and Measure

  2011

  (i) State and prove Fatou's lemma. State and prove Lebesgue's dominated convergence theorem. [You may assume the monotone convergence theorem.]

  In the rest of the question, let $f_{n}$ be a sequence of integrable functions on some measure space $(E, \mathcal{E}, \mu)$, and assume that $f_{n} \rightarrow f$ almost everywhere, where $f$ is a given integrable function. We also assume that $\int\left|f_{n}\right| d \mu \rightarrow \int|f| d \mu$ as $n \rightarrow \infty$.

  (ii) Show that $\int f_{n}^{+} d \mu \rightarrow \int f^{+} d \mu$ and that $\int f_{n}^{-} d \mu \rightarrow \int f^{-} d \mu$, where $\phi^{+}=\max (\phi, 0)$ and $\phi^{-}=\max (-\phi, 0)$ denote the positive and negative parts of a function $\phi$.

  (iii) Here we assume also that $f_{n} \geqslant 0$. Deduce that $\int\left|f-f_{n}\right| d \mu \rightarrow 0$.

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• Paper 1, Section II, I

  #Representation Theory

  2011

  Let $G$ be a finite group and $Z$ its centre. Suppose that $G$ has order $n$ and $Z$ has order $m$. Suppose that $\rho: G \rightarrow \mathrm{GL}(V)$ is a complex irreducible representation of degree $d .$

  (i) For $g \in Z$, show that $\rho(g)$ is a scalar multiple of the identity.

  (ii) Deduce that $d^{2} \leqslant n / m$.

  (iii) Show that, if $\rho$ is faithful, then $Z$ is cyclic.

  [Standard results may be quoted without proof, provided they are stated clearly.]

  Now let $G$ be a group of order 18 containing an elementary abelian subgroup $P$ of order 9 and an element $t$ of order 2 with $t x t^{-1}=x^{-1}$ for each $x \in P$. By considering the action of $P$ on an irreducible $\mathbb{C} G$-module prove that $G$ has no faithful irreducible complex representation.

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• Paper 2, Section II, I

  #Representation Theory

  2011

  State Maschke's Theorem for finite-dimensional complex representations of the finite group $G$. Show by means of an example that the requirement that $G$ be finite is indispensable.

  Now let $G$ be a (possibly infinite) group and let $H$ be a normal subgroup of finite index $r$ in $G$. Let $g_{1}, \ldots, g_{r}$ be representatives of the cosets of $H$ in $G$. Suppose that $V$ is a finite-dimensional completely reducible $\mathbb{C} G$-module. Show that

  (i) if $U$ is a $\mathbb{C} H$-submodule of $V$ and $g \in G$, then the set $g U=\{g u: u \in U\}$ is a $\mathbb{C} H$-submodule of $V$;

  (ii) if $U$ is a $\mathbb{C} H$-submodule of $V$, then $\sum_{i=1}^{r} g_{i} U$ is a $\mathbb{C} G$-submodule of $V$;

  (iii) $V$ is completely reducible regarded as a $\mathbb{C} H$-module.

  Hence deduce that if $\chi$ is an irreducible character of the finite group $G$ then all the constituents of $\chi_{H}$ have the same degree.

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• Paper 3, Section II, I

  #Representation Theory

  2011

  Define the character $\operatorname{Ind}_{H}^{G} \psi$ of a finite group $G$ which is induced by a character $\psi$ of a subgroup $H$ of $G$.

  State and prove the Frobenius reciprocity formula for the characters $\psi$ of $H$ and $\chi$ of $G$.

  Now suppose that $H$ has index 2 in $G$. An irreducible character $\psi$ of $H$ and an irreducible character $\chi$ of $G$ are said to be 'related' if

  $$\left\langle\operatorname{Ind}_{H}^{G} \psi, \chi\right\rangle_{G}=\left\langle\psi, \operatorname{Res}_{H}^{G} \chi\right\rangle_{H}>0$$

  Show that each $\psi$ of degree $d$ is either 'monogamous' in the sense that it is related to one $\chi$ (of degree $2 d$ ), or 'bigamous' in the sense that it is related to precisely two distinct characters $\chi_{1}, \chi_{2}$ (of degree $\left.d\right)$. Show that each $\chi$ is related to one bigamous $\psi$, or to two monogamous characters $\psi_{1}, \psi_{2}$ (of the same degree).

  Write down the degrees of the complex irreducible characters of the alternating group $A_{5}$. Find the degrees of the irreducible characters of a group $G$ containing $A_{5}$ as a subgroup of index 2 , distinguishing two possible cases.

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• Paper 4, Section II, I

  #Representation Theory

  2011

  Define the groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$.

  Show that $G=\mathrm{SU}(2)$ acts on the vector space of $2 \times 2$ complex matrices of the form

  $$V=\left\{A=\left(\begin{array}{cc} a & b \\ c & -a \end{array}\right) \in \mathrm{M}_{2}(\mathbb{C}): A+\overline{A^{t}}=0\right\}$$

  by conjugation. Denote the corresponding representation of $\mathrm{SU}(2)$ on $V$ by $\rho$.

  Prove the following assertions about this action:

  (i) The subspace $V$ is isomorphic to $\mathbb{R}^{3}$.

  (ii) The pairing $(A, B) \mapsto-\operatorname{tr}(A B)$ defines a positive definite non-degenerate $\mathrm{SU}(2)$ invariant bilinear form.

  (iii) The representation $\rho$ maps $G$ into $\mathrm{SO}(3)$. [You may assume that for any compact group $H$, and any $n \in \mathbb{N}$, there is a continuous group homomorphism $H \rightarrow \mathrm{O}(n)$ if and only if $H$ has an $n$-dimensional representation over $\mathbb{R}$.]

  Write down an orthonormal basis for $V$ and use it to show that $\rho$ is surjective with kernel $\{\pm I\}$.

  Use the isomorphism $\mathrm{SO}(3) \cong G /\{\pm I\}$ to write down a list of irreducible representations of $\mathrm{SO}(3)$ in terms of irreducibles for $\mathrm{SU}(2)$. [Detailed explanations are not required.]

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• Paper 1, Section II, 23G

  #Riemann Surfaces

  2011

  Suppose that $R_{1}$ and $R_{2}$ are Riemann surfaces, and $A$ is a discrete subset of $R_{1}$. For any continuous map $\alpha: R_{1} \rightarrow R_{2}$ which restricts to an analytic map of Riemann surfaces $R_{1} \backslash A \rightarrow R_{2}$, show that $\alpha$ is an analytic map.

  Suppose that $f$ is a non-constant analytic function on a Riemann surface $R$. Show that there is a discrete subset $A \subset R$ such that, for $P \in R \backslash A, f$ defines a local chart on some neighbourhood of $P$.

  Deduce that, if $\alpha: R_{1} \rightarrow R_{2}$ is a homeomorphism of Riemann surfaces and $f$ is a non-constant analytic function on $R_{2}$ for which the composite $f \circ \alpha$ is analytic on $R_{1}$, then $\alpha$ is a conformal equivalence. Give an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.

  [You may assume standard results for analytic functions on domains in the complex plane.]

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• Paper 2, Section II, 23G

  #Riemann Surfaces

  2011

  Let $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\tau$ is a fixed complex number with non-zero imaginary part. Suppose that $f$ is a meromorphic function on $\mathbb{C}$ for which the poles of $f$ are precisely the points in $\Lambda$, and for which $f(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. Assume moreover that $f^{\prime}(z)$ determines a doubly periodic function with respect to $\Lambda$ with $f^{\prime}(-z)=-f^{\prime}(z)$ for all $z \in \mathbb{C} \backslash \Lambda$. Prove that:

  (i) $f(-z)=f(z)$ for all $z \in \mathbb{C} \backslash \Lambda$.

  (ii) $f$ is doubly periodic with respect to $\Lambda$.

  (iii) If it exists, $f$ is uniquely determined by the above properties.

  (iv) For some complex number $A, f$ satisfies the differential equation $f^{\prime \prime}(z)=6 f(z)^{2}+A$.

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• Paper 3, Section II, G

  #Riemann Surfaces

  2011

  State the Classical Monodromy Theorem for analytic continuations in subdomains of the plane.

  Let $n, r$ be positive integers with $r>1$ and set $h(z)=z^{n}-1$. By removing $n$ semi-infinite rays from $\mathbb{C}$, find a subdomain $U \subset \mathbb{C}$ on which an analytic function $h^{1 / r}$ may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface $R$ for the complete analytic function $h^{1 / r}$.

  Let $Z$ denote the set of $n$th roots of unity and assume that the natural analytic covering map $\pi: R \rightarrow \mathbb{C} \backslash Z$ extends to an analytic map of Riemann surfaces $\tilde{\pi}: \tilde{R} \rightarrow \mathbb{C}_{\infty}$, where $\tilde{R}$ is a compactification of $R$ and $\mathbb{C}_{\infty}$ denotes the extended complex plane. Show that $\tilde{\pi}$ has precisely $n$ branch points if and only if $r$ divides $n$.

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• Paper 1, Section I, J

  #Statistical Modelling

  2011

  Let $Y_{1}, \ldots, Y_{n}$ be independent identically distributed random variables with model function $f(y, \theta), y \in \mathcal{Y}, \theta \in \Theta \subseteq \mathbb{R}$, and denote by $E_{\theta}$ and $\operatorname{Var}_{\theta}$ expectation and variance under $f(y, \theta)$, respectively. Define $U_{n}(\theta)=\sum_{i=1}^{n} \frac{\partial}{\partial \theta} \log f\left(Y_{i}, \theta\right)$. Prove that $E_{\theta} U_{n}(\theta)=0$. Show moreover that if $T=T\left(Y_{1}, \ldots, Y_{n}\right)$ is any unbiased estimator of $\theta$, then its variance satisfies $\operatorname{Var}_{\theta}(T) \geqslant\left(n \operatorname{Var}_{\theta}\left(U_{1}(\theta)\right)^{-1}\right.$. [You may use the Cauchy-Schwarz inequality without proof, and you may interchange differentiation and integration without justification if necessary.]

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• Paper 2, Section I, J

  #Statistical Modelling

  2011

  Let $f_{0}$ be a probability density function, with cumulant generating function $K$. Define what it means for a random variable $Y$ to have a model function of exponential dispersion family form, generated by $f_{0}$. Compute the cumulant generating function $K_{Y}$ of $Y$ and deduce expressions for the mean and variance of $Y$ that depend only on first and second derivatives of $K$.

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• Paper 3, Section I, J

  #Statistical Modelling

  2011

  Define a generalised linear model for a sample $Y_{1}, \ldots, Y_{n}$ of independent random variables. Define further the concept of the link function. Define the binomial regression model with logistic and probit link functions. Which of these is the canonical link function?

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• Paper 4, Section I, J

  #Statistical Modelling

  2011

  The numbers of ear infections observed among beach and non-beach (mostly pool) swimmers were recorded, along with explanatory variables: frequency, location, age, and sex. The data are aggregated by group, with a total of 24 groups defined by the explanatory variables.

  $$\begin{array}{ll} \text { freq } & \mathrm{F}=\text { frequent, } \mathrm{NF}=\text { infrequent } \\ \text { loc } & \mathrm{NB}=\text { non-beach, } \mathrm{B}=\text { beach } \\ \text { age } & 15-19,20-24,24-29 \\ \text { sex } & \mathrm{F}=\text { female, } \mathrm{M}=\text { male } \\ \text { count } & \text { the number of infections reported over a fixed time period } \\ \mathrm{n} & \text { the total number of swimmers } \end{array}$$

  The data look like this:

  $$\begin{array}{lrrrrrr} & \text { count } & \text { n } & \text { freq } & \text { loc } & \text { sex } & \text { age } \\ 1 & 68 & 31 & F & \text { NB } & \text { M } & 15-19 \\ 2 & 14 & 4 & F & \text { NB } & \text { F } & 15-19 \\ 3 & 35 & 12 & F & \text { NB } & \text { M } & 20-24 \\ 4 & 16 & 11 & F & \text { NB } & \text { F } & 20-24 \\ {[\ldots]} & & & & & & \\ 23 & 5 & 15 & \text { NF } & \text { B } & \text { M } & 25-29 \\ 24 & 6 & 6 & \text { NF } & \text { B } & \text { F } & 25-29 \end{array}$$

  Let $\mu_{j}$ denote the expected number of ear infections of a person in group $j$. Explain why it is reasonable to model count ${ }_{j}$ as Poisson with mean $n_{j} \mu_{j}$.

  We fit the following Poisson model:

  $$\log \left(\mathbb{E}\left(\operatorname{count}_{j}\right)\right)=\log \left(n_{j} \mu_{j}\right)=\log \left(n_{j}\right)+\mathbf{x}_{j} \beta$$

  where $\log \left(n_{j}\right)$ is an offset, i.e. an explanatory variable with known coefficient $1 .$ $\mathrm{R}$ produces the following (abbreviated) summary for the main effects model:

  

  Why are expressions freq $\mathrm{F}$, locB, age $15-19$, and sexF not listed?

  Suppose that we plan to observe a group of 20 female, non-frequent, beach swimmers, aged 20-24. Give an expression (using the coefficient estimates from the model fitted above) for the expected number of ear infections in this group.

  Now, suppose that we allow for interaction between variables age and sex. Give the $\mathrm{R}$ command for fitting this model. We test for the effect of this interaction by producing the following (abbreviated) ANOVA table:

  

  Briefly explain what test is performed, and what you would conclude from it. Does either of these models fit the data well?

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• Paper 1, Section II, J

  #Statistical Modelling

  2011

  The data consist of the record times in 1984 for 35 Scottish hill races. The columns list the record time in minutes, the distance in miles, and the total height gained during the route. The data are displayed in $R$ as follows (abbreviated):

  $\begin{array}{lrrr}\text { > hills } & & & \\ \text { Greenmantle } & 2.5 & 650 & 16.083 \\ \text { Carnethy } & 6.0 & 2500 & 48.350 \\ \text { Craig Dunain } & 6.0 & 900 & 33.650 \\ \text { Ben Rha } & 7.5 & 800 & 45.600 \\ \text { Ben Lomond } & 8.0 & 3070 & 62.267 \\ \text { [...] } & & & \\ \text { Cockleroi } & 4.5 & 850 & 28.100 \\ \text { Moffat Chase } & 20.0 & 5000 & 159.833\end{array}$

  Consider a simple linear regression of time on dist and climb. Write down this model mathematically, and explain any assumptions that you make. How would you instruct $R$ to fit this model and assign it to a variable hills. $\operatorname{lm} 1$ ?

  First, we test the hypothesis of no linear relationship to the variables dist and climb against the full model. $\mathrm{R}$ provides the following ANOVA summary:

  

  Using the information in this table, explain carefully how you would test this hypothesis. What do you conclude?

  The $\mathrm{R}$ command

  summary (hills. Im1)

  provides the following (slightly abbreviated) summary:

  

  Carefully explain the information that appears in each column of the table. What are your conclusions? In particular, how would you test for the significance of the variable climb in this model?

  

  Figure 1: Hills data: diagnostic plots

  Finally, we perform model diagnostics on the full model, by looking at studentised residuals versus fitted values, and the normal QQ-plot. The plots are displayed in Figure $1 .$ Comment on possible sources of model misspecification. Is it possible that the problem lies with the data? If so, what do you suggest?

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• Paper 4, Section II, J

  #Statistical Modelling

  2011

  Consider the general linear model $Y=X \beta+\epsilon$, where the $n \times p$ matrix $X$ has full rank $p \leqslant n$, and where $\epsilon$ has a multivariate normal distribution with mean zero and covariance matrix $\sigma^{2} I_{n}$. Write down the likelihood function for $\beta, \sigma^{2}$ and derive the maximum likelihood estimators $\hat{\beta}, \hat{\sigma}^{2}$ of $\beta, \sigma^{2}$. Find the distribution of $\hat{\beta}$. Show further that $\hat{\beta}$ and $\hat{\sigma}^{2}$ are independent.

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• Paper 1, Section II, D

  #Statistical Physics

  2011

  Describe the physical relevance of the microcanonical, canonical and grand canonical ensembles. Explain briefly the circumstances under which all ensembles are equivalent.

  The Gibbs entropy for a probability distribution $p(n)$ over states is

  $$S=-k_{B} \sum_{n} p(n) \log p(n) .$$

  By imposing suitable constraints on $p(n)$, show how maximising the entropy gives rise to the probability distributions for the microcanonical and canonical ensembles.

  A system consists of $N$ non-interacting particles fixed at points in a lattice. Each particle has three states with energies $E=-\epsilon, 0,+\epsilon$. If the system is at a fixed temperature $T$, determine the average energy $E$ and the heat capacity $C$. Evaluate each in the limits $T \rightarrow \infty$ and $T \rightarrow 0$.

  Describe a configuration of the system that would have negative temperature. Does this system obey the third law of thermodynamics?

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• Paper 2, Section II, D

  #Statistical Physics

  2011

  Write down the partition function for a single classical non-relativistic particle of mass $m$ moving in three dimensions in a potential $U(\mathbf{x})$ and in equilibrium with a heat bath at temperature $T$.

  A system of $N$ non-interacting classical non-relativistic particles, in equilibrium at temperature $T$, is placed in a potential

  $$U(\mathbf{x})=\frac{\left(x^{2}+y^{2}+z^{2}\right)^{n}}{V^{2 n / 3}}$$

  where $n$ is a positive integer. Using the partition function, show that the free energy is

  $$F=-N k_{B} T\left(\log V+\frac{3}{2} \frac{n+1}{n} \log k_{B} T+\log I_{n}+\text { const }\right)$$

  where

  $$I_{n}=\left(\frac{m}{2 \pi \hbar^{2}}\right)^{3 / 2} \int_{0}^{\infty} 4 \pi u^{2} \exp \left(-u^{2 n}\right) d u$$

  Explain the physical relevance of the constant term in the expression $(*)$.

  Viewing $V$ as an external parameter, akin to volume, compute the conjugate pressure $p$ and show that the equation of state coincides with that of an ideal gas.

  Compute the energy $E$, heat capacity $C_{V}$ and entropy $S$ of the gas. Determine the local particle number density as a function of $|\mathbf{x}|$.

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• Paper 3, Section II, D

  #Statistical Physics

  2011

  A gas of non-interacting particles has energy-momentum relationship $E=A(\hbar k)^{\alpha}$ for some constants $A$ and $\alpha$. Determine the density of states $g(E) d E$ in a threedimensional volume $V$.

  Explain why the chemical potential $\mu$ satisfies $\mu<0$ for the Bose-Einstein distribution.

  Show that an ideal quantum Bose gas with the energy-momentum relationship above has

  $$p V=\frac{\alpha E}{3} .$$

  If the particles are bosons at fixed temperature $T$ and chemical potential $\mu$, write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of $\alpha$ for which there exists a Bose-Einstein condensate at sufficiently low temperatures.

  Discuss whether a gas of photons can undergo Bose-Einstein condensation.

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• Paper 4, Section II, D

  #Statistical Physics

  2011

  (i) Define the Gibbs free energy for a gas of $N$ particles with pressure $p$ at a temperature $T$. Explain why it is necessarily proportional to the number of particles $N$ in the system. Given volume $V$ and chemical potential $\mu$, prove that

  $$\left.\frac{\partial \mu}{\partial p}\right|_{T}=\frac{V}{N} .$$

  (ii) The van der Waals equation of state is

  $$\left(p+\frac{a N^{2}}{V^{2}}\right)(V-N b)=N k_{B} T$$

  Explain the physical significance of the terms with constants $a$ and $b$. Sketch the isotherms of the van der Waals equation. Show that the critical point lies at

  $$k_{B} T_{c}=\frac{8 a}{27 b}, \quad V_{c}=3 b N, \quad p_{c}=\frac{a}{27 b^{2}} .$$

  (iii) Describe the Maxwell construction to determine the condition for phase equilibrium. Hence sketch the regions of the van der Waals isotherm at $T<T_{c}$ that correspond to metastable and unstable states. Sketch those regions that correspond to stable liquids and stable gases.

  (iv) Show that, as the critical point is approached along the co-existence curve,

  $$V_{\text {gas }}-V_{\text {liquid }} \sim\left(T_{c}-T\right)^{1 / 2} .$$

  Show that, as the critical point is approached along an isotherm,

  $$p-p_{c} \sim\left(V-V_{c}\right)^{3} .$$

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• Paper 1, Section II, J

  #Stochastic Financial Models

  2011

  In a one-period market, there are $n$ assets whose prices at time $t$ are given by $S_{t}=\left(S_{t}^{1}, \ldots, S_{t}^{n}\right)^{T}, t=0,1$. The prices $S_{1}$ of the assets at time 1 have a $N(\mu, V)$ distribution, with non-singular covariance $V$, and the prices $S_{0}$ at time 0 are known constants. In addition, there is a bank account giving interest $r$, so that one unit of cash invested at time 0 will be worth $(1+r)$ units of cash at time 1 .

  An agent with initial wealth $w_{0}$ chooses a portfolio $\theta=\left(\theta^{1}, \ldots, \theta^{n}\right)$ of the assets to hold, leaving him with $x=w_{0}-\theta \cdot S_{0}$ in the bank account. His objective is to maximize his expected utility

  $$E\left(-\exp \left[-\gamma\left\{x(1+r)+\theta \cdot S_{1}\right\}\right]\right) \quad(\gamma>0)$$

  Find his optimal portfolio in each of the following three situations:

  (i) $\theta$ is unrestricted;

  (ii) no investment in the bank account is allowed: $x=0$;

  (iii) the initial holdings $x$ of cash must be non-negative.

  For the third problem, show that the optimal initial holdings of cash will be zero if and only if

  $$\frac{S_{0} \cdot(\gamma V)^{-1} \mu-w_{0}}{S_{0} \cdot(\gamma V)^{-1} S_{0}} \geqslant 1+r$$

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• Paper 2, Section II, J

  #Stochastic Financial Models

  2011

  Consider a symmetric simple random walk $\left(Z_{n}\right)_{n \in \mathbb{Z}^{+}}$taking values in statespace $I=h \mathbb{Z}^{2} \equiv\{(i h, j h): i, j \in \mathbb{Z}\}$, where $h \equiv N^{-1}(N$ an integer $)$. Writing $Z_{n} \equiv\left(X_{n}, Y_{n}\right)$, the transition probabilities are given by

  $$P\left(\Delta Z_{n}=(h, 0)\right)=P\left(\Delta Z_{n}=(0, h)\right)=P\left(\Delta Z_{n}=(-h, 0)\right)=P\left(\Delta Z_{n}=(0,-h)\right)=\frac{1}{4},$$

  where $\Delta Z_{n} \equiv Z_{n}-Z_{n-1}$.

  What does it mean to say that $\left(M_{n}, \mathcal{F}_{n}\right)_{n \in \mathbb{Z}^{+}}$is a martingale? Find a condition on $\theta$ and $\lambda$ such that

  $$M_{n}=\exp \left(\theta X_{n}-\lambda Y_{n}\right)$$

  is a martingale. If $\theta=i \alpha$ for some real $\alpha$, show that $M$ is a martingale if

  $$e^{-\lambda h}=2-\cos (\alpha h)-\sqrt{(2-\cos (\alpha h))^{2}-1}$$

  Suppose that the random walk $Z$ starts at position $(0,1) \equiv(0, N h)$ at time 0 , and suppose that

  $$\tau=\inf \left\{n: Y_{n}=0\right\} .$$

  Stating fully any results to which you appeal, prove that

  $$E \exp \left(i \alpha X_{\tau}\right)=e^{-\lambda}$$

  where $\lambda$ is as given at $(*)$. Deduce that as $N \rightarrow \infty$

  $$E \exp \left(i \alpha X_{\tau}\right) \rightarrow e^{-|\alpha|}$$

  and comment briefly on this result.

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• Paper 3, Section II, J

  #Stochastic Financial Models

  2011

  First, what is a Brownian motion?

  (i) The price $S_{t}$ of an asset evolving in continuous time is represented as

  $$S_{t}=S_{0} \exp \left(\sigma W_{t}+\mu t\right)$$

  where $W$ is a standard Brownian motion, and $\sigma$ and $\mu$ are constants. If riskless investment in a bank account returns a continuously-compounded rate of interest $r$, derive a formula for the time-0 price of a European call option on the asset $S$ with strike $K$ and expiry $T$. You may use any general results, but should state them clearly.

  (ii) In the same financial market, consider now a derivative which pays

  $$Y=\left\{\exp \left(T^{-1} \int_{0}^{T} \log \left(S_{u}\right) d u\right)-K\right\}^{+}$$

  at time $T$. Find the time-0 price for this derivative. Show that it is less than the price of the European call option which you derived in (i).

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• Paper 4, Section II, J

  #Stochastic Financial Models

  2011

  In a two-period model, two agents enter a negotiation at time 0 . Agent $j$ knows that he will receive a random payment $X_{j}$ at time $1(j=1,2)$, where the joint distribution of $\left(X_{1}, X_{2}\right)$ is known to both agents, and $X_{1}+X_{2}>0$. At the outcome of the negotiation, there will be an agreed risk transfer random variable $Y$ which agent 1 will pay to agent 2 at time 1 . The objective of agent 1 is to maximize $E U_{1}\left(X_{1}-Y\right)$, and the objective of agent 2 is to maximize $E U_{2}\left(X_{2}+Y\right)$, where the functions $U_{j}$ are strictly increasing, strictly concave, $C^{2}$, and have the properties that

  $$\lim _{x \downarrow 0} U_{j}^{\prime}(x)=+\infty, \quad \lim _{x \uparrow \infty} U_{j}^{\prime}(x)=0$$

  Show that, unless there exists some $\lambda \in(0, \infty)$ such that

  $$\frac{U_{1}^{\prime}\left(X_{1}-Y\right)}{U_{2}^{\prime}\left(X_{2}+Y\right)}=\lambda \quad \text { almost surely }$$

  the risk transfer $Y$ could be altered to the benefit of both agents, and so would not be the conclusion of the negotiation.

  Show that, for given $\lambda>0$, the relation $(*)$ determines a unique risk transfer $Y=Y_{\lambda}$, and that $X_{2}+Y_{\lambda}$ is a function of $X_{1}+X_{2}$.

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• Paper 1, Section I, F

  #Topics in Analysis

  2011

  (i) State the Baire Category Theorem for metric spaces in its closed sets version.

  (ii) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a complex analytic function which is not a polynomial. Prove that there exists a point $z_{0} \in \mathbb{C}$ such that each coefficient of the Taylor series of $f$ at $z_{0}$ is non-zero.

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• Paper 2, Section I, F

  #Topics in Analysis

  2011

  (i) Let $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ be any set of $n$ distinct numbers. Show that there exist numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the formula

  $$\int_{-1}^{1} p(x) d x=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

  is valid for every polynomial $p$ of degree $\leqslant n-1$.

  (ii) For $n=0,1,2, \ldots$, let $p_{n}$ be the Legendre polynomial, over $[-1,1]$, of degree $n$. Suppose that $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ are the roots of $p_{n}$, and $A_{1}, A_{2}, \ldots, A_{n}$ are the numbers corresponding to $x_{1}, x_{2}, \ldots, x_{n}$ as in (i).

  [You may assume without proof that for $n \geqslant 1, p_{n}$ has $n$ distinct roots in $[-1,1] .$ ]

  Prove that the integration formula in (i) is now valid for any polynomial $p$ of degree $\leqslant 2 n-1$.

  (iii) Is it possible to choose $n$ distinct points $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ and corresponding numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the integration formula in (i) is valid for any polynomial $p$ of degree $\leqslant 2 n$ ? Justify your answer.

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• Paper 3, Section I, F

  #Topics in Analysis

  2011

  Let $\Gamma=\{z \in \mathbb{C}: z \neq 1,|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=1\}$.

  (i) Prove that, for any $\zeta \in \mathbb{C}$ with $|\operatorname{Re}(\zeta)|+|\operatorname{Im}(\zeta)|>1$ and any $\epsilon>0$, there exists a complex polynomial $p$ such that

  $$\sup _{z \in \Gamma}\left|p(z)-(z-\zeta)^{-1}\right|<\epsilon$$

  (ii) Does there exist a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow(z-1)^{-1}$ for every $z \in \Gamma ?$ Justify your answer.

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• Paper 4, Section I, $2 F$

  #Topics in Analysis

  2011

  (a) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin. State precisely a theorem about homotopy invariance of the winding number.

  (b) Let $B=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and let $f: B \rightarrow \mathbb{C}$ be a continuous map satisfying

  $$|f(z)-z| \leqslant 1$$

  for each $z \in \partial B$.

  (i) For $0 \leqslant t \leqslant 1$, let $\gamma(t)=f\left(e^{2 \pi i t}\right)$. If $\gamma(t) \neq 0$ for each $t \in[0,1]$, prove that $w(\gamma ; 0)=1$.

  [Hint: Consider a suitable homotopy between $\gamma$ and the map $\gamma_{1}(t)=e^{2 \pi i t}$, $0 \leqslant t \leqslant 1 .]$

  (ii) Deduce that $f(z)=0$ for some $z \in B$.

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• Paper 2, Section II, F

  #Topics in Analysis

  2011

  Let $C[0,1]$ be the space of real continuous functions on the interval $[0,1]$. A mapping $L: C[0,1] \rightarrow C[0,1]$ is said to be positive if $L(f) \geqslant 0$ for each $f \in C[0,1]$ with $f \geqslant 0$, and linear if $L(a f+b g)=a L(f)+b L(g)$ for all functions $f, g \in C[0,1]$ and constants $a, b \in \mathbb{R}$.

  (i) Let $L_{n}: C[0,1] \rightarrow C[0,1]$ be a sequence of positive, linear mappings such that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for the three functions $f(x)=1, x, x^{2}$. Prove that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for every $f \in C[0,1]$.

  (ii) Define $B_{n}: C[0,1] \rightarrow C[0,1]$ by

  $$B_{n}(f)(x)=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}$$

  where $\left(\begin{array}{l}n \\ k\end{array}\right)=\frac{n !}{k !(n-k) !}$. Using the result of part (i), or otherwise, prove that $B_{n}(f) \rightarrow f$ uniformly on $[0,1]$.

  (iii) Let $f \in C[0,1]$ and suppose that

  $$\int_{0}^{1} f(x) x^{4 n} d x=0$$

  for each $n=0,1, \ldots$ Prove that $f$ must be the zero function.

  [You should not use the Stone-Weierstrass theorem without proof.]

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• Paper 3, Section II, F

  #Topics in Analysis

  2011

  Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous and let $n$ be a positive integer. For $g:[0,1] \rightarrow \mathbb{R}$ a continuous function, write $\|f-g\|_{L^{\infty}}=\sup _{x \in[0,1]}|f(x)-g(x)|$.

  (i) Let $p$ be a polynomial of degree at most $n$ with the property that there are $(n+2)$ distinct points $x_{1}, x_{2}, \ldots, x_{n+2} \in[0,1]$ with $x_{1}<x_{2}<\ldots<x_{n+2}$ such that

  $$f\left(x_{j}\right)-p\left(x_{j}\right)=(-1)^{j}\|f-p\|_{L^{\infty}}$$

  for each $j=1,2, \ldots, n+2$. Prove that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  (ii) Prove that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  [If you deduce this from a more general result about abstract normed spaces, you must prove that result.]

  (iii) Let $Y=\left\{y_{1}, y_{2}, \ldots, y_{n+2}\right\}$ be any set of $(n+2)$ distinct points in $[0,1]$.

  (a) For $j=1,2, \ldots, n+2$, let

  $$r_{j}(x)=\prod_{k=1, k \neq j}^{n+2} \frac{x-y_{k}}{y_{j}-y_{k}}$$

  $t(x)=\sum_{j=1}^{n+2} f\left(y_{j}\right) r_{j}(x)$ and $r(x)=\sum_{j=1}^{n+2}(-1)^{j} r_{j}(x)$. Explain why there is a unique number $\lambda \in \mathbb{R}$ such that the degree of the polynomial $t-\lambda r$ is at $\operatorname{most} n$.

  (b) Let $\|f-g\|_{L^{\infty}(Y)}=\sup _{x \in Y}|f(x)-g(x)|$. Deduce from part (a) that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}(Y)} \leqslant\|f-q\|_{L^{\infty}(Y)}$$

  for every polynomial $q$ of degree at most $n$.

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• Paper 1, Section II, 39B

  #Waves

  2011

  An inviscid fluid with sound speed $c_{0}$ occupies the region $0<y<\pi \alpha, 0<z<\pi \beta$ enclosed by the rigid boundaries of a rectangular waveguide. Starting with the acoustic wave equation, find the dispersion relation $\omega(k)$ for the propagation of sound waves in the $x$-direction.

  Hence find the phase speed $c(k)$ and the group velocity $c_{g}(k)$ of both the dispersive modes and the nondispersive mode, and sketch the form of the results for $k, \omega>0$.

  Define the time and cross-sectional average appropriate for a mode with frequency $\omega$. For each dispersive mode, show that the average kinetic energy is equal to the average compressive energy.

  A general multimode acoustic disturbance is created within the waveguide at $t=0$ in a region around $x=0$. Explain briefly how the amplitude of the disturbance varies with time as $t \rightarrow \infty$ at the moving position $x=V t$ for each of the cases $0<V<c_{0}$, $V=c_{0}$ and $V>c_{0}$. [You may quote without proof any generic results from the method of stationary phase.]

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• Paper 2, Section II, 38B

  #Waves

  2011

  A uniform elastic solid with wavespeeds $c_{P}$ and $c_{S}$ occupies the region $z<0$. An $S$-wave with displacement

  $$\mathbf{u}=(\cos \theta, 0,-\sin \theta) \mathrm{e}^{\mathrm{i} k(x \sin \theta+z \cos \theta)-\mathrm{i} \omega t}$$

  is incident from $z<0$ on a rigid boundary at $z=0$. Find the form and amplitudes of the reflected waves.

  When is the reflected $P$-wave evanescent? Show that if the $P$-wave is evanescent then the amplitude of the reflected $S$-wave has the same magnitude as the incident wave, and interpret this result physically.

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• Paper 3, Section II, 38B

  #Waves

  2011

  The dispersion relation in a stationary medium is given by $\omega=\Omega_{0}(\mathbf{k})$, where $\Omega_{0}$ is a known function. Show that, in the frame of reference where the medium has a uniform velocity $-\mathbf{U}$, the dispersion relation is given by $\omega=\Omega_{0}(\mathbf{k})-\mathbf{U} \cdot \mathbf{k}$.

  An aircraft flies in a straight line with constant speed $M c_{0}$ through air with sound speed $c_{0}$. If $M>1$ show that, in the reference frame of the aircraft, the steady waves lie behind it on a cone of semi-angle $\sin ^{-1}(1 / M)$. Show further that the unsteady waves are confined to the interior of the cone.

  A small insect swims with constant velocity $\mathbf{U}=(U, 0)$ over the surface of a pool of water. The resultant capillary waves have dispersion relation $\omega^{2}=T|\mathbf{k}|^{3} / \rho$ on stationary water, where $T$ and $\rho$ are constants. Show that, in the reference frame of the insect, steady waves have group velocity

  $$\mathbf{c}_{g}=U\left(\frac{3}{2} \cos ^{2} \beta-1, \frac{3}{2} \cos \beta \sin \beta\right),$$

  where $\mathbf{k} \propto(\cos \beta, \sin \beta)$. Deduce that the steady wavefield extends in all directions around the insect.

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• Paper 4, Section II, 38B

  #Waves

  2011

  Show that, in the standard notation for one-dimensional flow of a perfect gas, the Riemann invariants $u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant on characteristics $C_{\pm}$given by

  $$\frac{\mathrm{d} x}{\mathrm{~d} t}=u \pm c$$

  Such a gas occupies the region $x>X(t)$ in a semi-infinite tube to the right of a piston at $x=X(t)$. At time $t=0$, the piston and the gas are at rest, $X=0$, and the gas is uniform with $c=c_{0}$. For $t>0$ the piston accelerates smoothly in the positive $x$-direction. Show that, prior to the formation of a shock, the motion of the gas is given parametrically by

  $$u(x, t)=\dot{X}(\tau) \quad \text { on } \quad x=X(\tau)+\left[c_{0}+\frac{1}{2}(\gamma+1) \dot{X}(\tau)\right](t-\tau)$$

  in a region that should be specified.

  For the case $X(t)=\frac{2}{3} c_{0} t^{3} / T^{2}$, where $T>0$ is a constant, show that a shock first forms in the gas when

  $$t=\frac{T}{\gamma+1}(3 \gamma+1)^{1 / 2}$$

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• Paper 4 , Section II, D

  #Variational Principles

  2011

  Derive the Euler-Lagrange equation for the integral

  $$\int_{x_{0}}^{x_{1}} f\left(y, y^{\prime}, y^{\prime \prime}, x\right) d x$$

  where the endpoints are fixed, and $y(x)$ and $y^{\prime}(x)$ take given values at the endpoints.

  Show that the only function $y(x)$ with $y(0)=1, y^{\prime}(0)=2$ and $y(x) \rightarrow 0$ as $x \rightarrow \infty$ for which the integral

  $$\int_{0}^{\infty}\left(y^{2}+\left(y^{\prime}\right)^{2}+\left(y^{\prime}+y^{\prime \prime}\right)^{2}\right) d x$$

  is stationary is $(3 x+1) e^{-x}$.

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